Mathematics

9

Mathematics

ing

Number

Reason

Proble

m Sol

ving

Algebra

Statistics

Measurement

National Curriculum

tanding

Probability

Unders

Fluen

cy

Geometry

Dr Terry Dwyer

Mathematics National Curriculum

9

Dr Terry Dwyer CertT, BAppSc, BEd, GDipEd, MEd(Hons), PhD Head of Mathematics

www.drdwyer.com.au

The cover: Maths is the pathway to success in our technological economy and society.

“In today’s world, economic access and full citizenship depend crucially on math and science literacy.” - Robert P. Moses. “Mathematics makes a significant contribution to our modern society; its basic skills are vital for the life opportunities of the youth. Let us all embrace and encourage the study of this indispensable subject.“ - Alfred L. Teye.

Dr Dwyer Pty Ltd ABN 27105593922 10 Moss Court Stanthorpe 4380 www.drdwyer.com.au Copyright © Dr Dwyer Pty Ltd First Published 2011 COPYRIGHT Apart from fair dealing for the purposes of study, research, criticism or review, or as permitted under Part VB of the Copyright Act, no part of this book may be reproduced by any process without permission. National Library of Australia Cataloguing-in-Publication entry Dwyer, T. (Terry), Mathematics 9 : National Curriculum 1st ed. Includes index. For secondary school year age. ISBN: 978-0-646-55957-5 1. Mathematics--Problems, exercises etc. 510

Published in Australia by Dr Dwyer Pty Ltd Printed in Malaysia

Contents - Overview

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5

Indices 1 Algebra 1 Area Linear & Non-linear Graphs Review 1

1 17 33 49 65



Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10

Proportion Pythagoras' Theorem Geometry Statistics Review 2

73 89 103 119 135



Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15

Indices 2 Trigonometry 1 Volume Probability Review 3

143 159 175 189 205



Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20

Coordinate Geometry Trigonometry 2 Algebra 2 Data Review 4

213 229 245 261 275

Glossary Answers Index

285 295 311

Chapter 1 Indices 1 1 A Task 1 History 1 WarmUp 2 Indices 3 Index Law 1 4 Index Law 2 4 Index Law 3 5 Index Law 4 5 Index Law 5 6 Summary 8 Mental Computation 9 NAPLAN Questions 10 Competition Questions 11 Investigations 12 A Couple of Puzzles 13 A Game 13 A Sweet Trick 13 Technology 14 Chapter Review 1 15 Chapter Review 2 16

Chapter 3 Area 33 A Task 33 History 33 Area Warmup 34 Composite Shapes 35 Prisms 36 Surface Area 37 Cylinders 38 Circle Review 38 Cylinder Surface Area 39 All Together 40 Mental Computation 41 NAPLAN Questions 42 Competition Questions 43 Investigations 44 Technology 45 A Couple of Puzzles 46 A Game 46 A Sweet Trick 46 Chapter Review 1 47 Chapter Review 2 48 History 49

Chapter 2 Algebra 1 17 A Task 17 History 17 Algebra Warmup 18 Distributive Law 20 Factorisation 23 Mental Computation 25 NAPLAN Questions 26 Competition Questions 27 Investigations 28 A Couple of Puzzles 29 A Game 29 A Sweet Trick 29 Technology 30 Chapter Review 1 31 Chapter Review 2 32

Chapter 4 Linear & Non-linear 49 A Task 49 Linear Rules 50 Sketching Linear Graphs 53 Linear Graphs 54 Non-Linear Graphs 55 Mental Computation 57 NAPLAN Questions 58 Competition Questions 59 Investigations 60 Technology 61 A Couple of Puzzles 62 A Game 62 A Sweet Trick 62 Chapter Review 1 63 Chapter Review 2 64 Chapter 5 Review 1 Review 1 Review 2

65 66 70

Chapter 6 Proportion 73 A Task 73 History 73 Warmup 74 Proportion 76 Direct Proportion 78 Inverse Proportion 80 Money and Proportion 81 Mental Computation 82 Competition Questions 83 Investigations 84 A Couple of Puzzles 85 A Game 85 A Sweet Trick 85 Technology 86 Chapter Review 1 87 Chapter Review 2 88 History 89

Chapter 8 Geometry 103 A Task 103 History 103 Congruent Triangles 104 Tests for Congruent Triangles 105 Similarity Transformation 106 Similar Triangles 108 Tests for Similar Triangles 109 Mental Computation 112 Competition Questions 113 Technology 114 A Couple of Puzzles 116 A Game 116 A Sweet Trick 116 Chapter Review 1 117 Chapter Review 2 118 History 119

Chapter 7 Pythagoras' Theorem 89 A Task 89 Pythagorean Triads 90 Pythagoras' Theorem 91 Hypotenuse 92 The Shorter Sides 93 Length of a line 94 Our Number System 95 Mental Computation 96 Competition Questions 97 Investigations 98 Technology 99 A Couple of Puzzles 100 A Game 100 A Sweet Trick 100 Chapter Review 1 101 Chapter Review 2 102

Chapter 9 Statistics 119 A Task 119 Descriptive Statistics 120 Stem-and-Leaf Plots 122 Histograms 124 Comparative Analysis 126 Mental Computation 128 Competition Questions 129 Investigations 130 A Couple of Puzzles 131 A Game 131 A Sweet Trick 131 Technology 132 Chapter Review 1 133 Chapter Review 2 134

Chapter 10 Review 2 Review 1 Review 2

135 136 139

Chapter 11 Indices 2 143 A Task 143 History 143 Warmup 144 Index Law 1 145 Index Law 2 145 Index Law 3 146 Index Law 4 146 Index Law 5 147 Scientific Notation 148 Mental Computation 152 Competition Questions 153 Investigations 154 A Couple of Puzzles 155 A Game 155 A Sweet Trick 155 Technology 156 Chapter Review 1 157 Chapter Review 2 158

Chapter 13 Volume 175 A Task 175 History 175 Area Warmup 176 Composite Shapes 177 Prisms 178 Units of Volume 178 Volume of Prisms 179 Composite Solids 180 Practical Applications 181 Mental Computation 182 Competition Questions 183 Investigations 184 A Couple of Puzzles 185 A Game 185 A Sweet Trick 185 Technology 186 Chapter Review 1 187 Chapter Review 2 188

Chapter 12 Trigonometry 1 159 A Task 159 History 159 Pythagoras' Theorem 160 Naming Sides 162 Trigonometry 163 The Tan Ratio 164 Mental Computation 168 Competition Questions 169 A Couple of Puzzles 170 A Game 170 A Sweet Trick 170 Investigations 171 Technology 172 Chapter Review 1 173 Chapter Review 2 174

Chapter 14 Probability 189 A Task 189 History 189 Warm Up 190 Theoretical Probability 191 Experimental Probability 192 Venn Diagrams 197 Mental Computation 198 Competition Questions 199 Investigations 200 Technology 201 A Couple of Puzzles 202 A Game 202 A Sweet Trick 202 Chapter Review 1 203 Chapter Review 2 204 Chapter 15 Review 3 Review 1 Review 2

205 206 209

Chapter 16 Coordinate Geometry 213 A Task 213 History 213 Distance Between 2 Points 214 Midpoint 216 Gradient 218 Formulas 220 Mental Computation 222 Competition Questions 223 Investigations 224 A Couple of Puzzles 225 A Game 225 A Sweet Trick 225 Technology 226 Chapter Review 1 227 Chapter Review 2 228

Chapter 18 Algebra 2 245 A Task 245 History 245 Integer Warmup 246 Index Law Warmup 247 Algebra Warmup 248 Distributive Law 250 Factorisation 251 Mental Computation 254 Competition Questions 255 Investigations 256 A Couple of Puzzles 257 A Game 257 A Sweet Trick 257 Technology 258 Chapter Review 1 259 Chapter Review 2 260

Chapter 17 Trigonometry 2 229 A Task 229 History 229 Pythagoras' Theorem 230 The Tan Ratio 231 The Sine Ratio 232 The Cos Ratio 233 Trigonometry 234 Mental Computation 238 Competition Questions 239 A Couple of Puzzles 240 A Game 240 A Sweet Trick 240 Investigations 241 Technology 242 Chapter Review 1 243 Chapter Review 2 244

Chapter 19 Data 261 A Task 261 History 261 Data 262 Collecting Data 263 Sampling 264 Stratified Sampling 265 Questionnaires 266 Mental Computation 268 Competition Questions 269 Investigations 270 Technology 271 A Couple of Puzzles 272 A Game 272 A Sweet Trick 272 Chapter Review 1 273 Chapter Review 2 274 Paradoxes 283 Isometric Drawing 284 Chapter 20 Review 4 Review 1 Review 2

275 276 279

Glossary 285 Answers 295 Index 311

Preface

This text has been written for Year 9 students. The aim of the text is to assist students in investigating and understanding the exciting and very important world of Mathematics and to implement the intent of the Australian Mathematics Curriculum. A literature review of learning from school textbooks was used to enhance the format of this textbook.

Each chapter, apart from Review, contains:              

Numerous worked examples Numerous sets of graded exercises An open-ended rich task Mental computation Technology in mathematics Investigations Puzzles NAPLAN questions Maths competition preparation A mathematics game A mathematics trick A bit of mathematics history Careers using mathematics Chapter review

Acknowledgments

A heart-felt thank you to my wife Karen for your encouragement, advice, text design, images, illustrations, and above all, your loving support. Public Domain Images: pp. 17,40,48,73,143,175,189,213,229,245,261. Public Domain Images courtesy of nasaimages.org: pp. 151,156.

Resources www.drdwyer.com.au

Number and Algebra → Real Numbers  Apply index laws to numerical expressions with integer indices.  Connect different strategies for simplifying expressions with indices to illustrate the meaning of negative indices.  Move fluently between representations of numeric and algebraic terms with negative indices.  Apply knowledge of index laws to algebraic terms and simplify algebraic expressions, using both positive and negative integral indices.

Infinite, adj. 1. exceedingly great, unlimited, immeasurably large.

A TASK • Everyone should know about infinity. • Every child learns to count to ten, then to twenty, then to one hundred. • Then what? Infinity of course. Research infinity and then give a five minute explanation of infinity to the rest of the class (Try to include one of the interesting stories about infinity).

A LITTLE BIT OF HISTORY • The Romans used "Decies centena milia (ten hundred thousand) as the Roman words for 1 000 000. • The French, in the 13th century, were the first to use the word 1 million. • The Indians were the most advanced with large numbers and had by the 7th century defined infinity as having a denominator of zero. 1 = 10 0.1

1 = 1000 0.001

A 1 m cubic block will have 1 million 1 cm cubes.

1 =∞ 0

1 = 1000000 0.000001

1

WarmUp Exercise 1.1 Calculate each of the following: 1 1×1 2 2×2 5 10×10 6 1×1×1×1 9 4×4×4 10 5×5×5 13 3×3×3×3 14 10×10×10×10

3 3×3 7 2×2×2 11 10×10×10 15 5×5×5×5

4 4×4 8 3×3×3 12 2×2×2×2

16 Copy and complete the following table: Squares are often used. (Area square = side2)

Square

Cubes happen now and then. (Volume cube = side3)

Cube

Fourth

Fifth

1 2 3 4 5 17 10×10 19 10×10×10×10 21 10×10×10×10×10×10

18 10×10×10 20 10×10×10×10×10 22 10×10×10×10×10×10×10

Exercise 1.2 Use a calculator to calculate each of the following: 5×5×5×5×5×5×5x5 On a calculator:

2

On some calculators, the power button is yx.

5 ^ 8 = 390625

1 2×2×2×2×2×2

2 5×5×5×5×5×5×5

3 1×1×1×1×1×1×1×1

4 10×10×10×10×10×10

5 2×2×2×2×2×2×2×2

6 22×22×22×22

7 (22)4

8

9 3×3×3×3×3×3×3×3×3

10 33×33×33

11 (33)3

12

22 33

4

3

yx

Indices A convenient way of writing 2×2×2 is

3 2

Exercise 1.3 Write the following in index form: 2×2×2×2×2 a×a×a×a = 25



1 4×4×4 4 10×10×10 7 m×m×m×m×m

Index Base

Indices save a lot of effort.

= a4



2 2×2×2×2 5 b×b×b×b×b 8 9×9×9×9

3 a×a×a 6 h×h×h 9 3×3×3×3×3×3

Exercise 1.4 Write the following in factor form: 34 b3 = 3×3×3×3 = b×b×b 1 43 4 27 7 x4

2 b4 5 62 8 p5

3 52 6 m5 9 14

Exercise 1.5 Write the following in index form: 2×2×2×4×4 abbaaab

= 23×42

1 aabbbaa 4 2×3×2×3×2×3×2 7 ppqrppqqrrrp



= a4×b3

2 3×3×3×2×2 3 abaaababb 5 bggggbbbg 6 zzzzzzzzzzz 8 2×2×2×3×4×4×4×3 9 4gg4g4g4gg

Biochemists study the chemistry of living things in the fields of medicine,

agriculture, the environment, and manufacturing. • Relevant school subjects are English, Mathematics, Chemsitry, Biology. • Courses normally involve a Degree with a major in chemistry/biochemistry.

Chapter 1 Indices 1

3

Index Law 1 Multiplying Indices:

Multiplying Indices: 24×22 = 2×2×2×2 × 2×2 = 26

or

Index Law 1 am×an = am+n

24×22 = 24+2 = 26

Exercise 1.6 Simplify and write the following in index form: 23×22 = 2×2×2 × 2×2 = 25 a2×a5 = a×a × a×a×a×a×a = a7 or 23×22 = 23+2 = 25 1 22×23 5 a2×a2 9 23×21 13 c4×c3 17 2.45×2.42 21 22×23×22

or a2×a5

2 33×32 6 b3×b2 10 t2×t5 14 102×104 18 z3×z2 22 34×33×32

= a2+5 = a7

3 24×22 7 z5×z3 11 52×53 15 a2×a3 19 0.55×0.54 23 a2×a3×a3

4 44×43 8 w4×w3 12 104×105 16 b5×b 20 105×102 24 u2×u2×u3

Index Law 2 Dividing Indices: a3÷a2 =

a ×a ×a a ×a

or

=a

Index Law 2

Dividing Indices: a3÷a2

am÷an = am−n

= a3−2 = a

Exercise 1.7 Simplify and write the following in index form: 23÷22 =

2× 2× 2 2× 2

a6÷a2 =

=2

or a6÷a2

or 23÷22 = 23−2 = 2 1 25÷23 5 a6÷a2 9 23÷21 13 a4÷a3 17 19

4

m4



m2 c

7

c

4



2 45÷42 6 b7÷b2 10 t7÷t4 14 1.24÷1.22 18 20

a ×a ×a ×a ×a ×a a ×a

= a×a×a×a = a4

= a6−2 = a4

3 24÷22 7 105÷103 11 56÷54 15 29×29÷23

4 44÷43 8 w4÷w3 12 105÷102 16 p3÷p3

e5 e2 10

4

102

m4÷m2 =

m4 m2

They are the same thing.

Index Law 3 Power Indices: (23)2 = (2×2×2)2 = (2×2×2)×(2×2×2) = 26

Power Indices: (23)2 = 23×2 = 26

or

Exercise 1.8 Simplify and write the following in index form: (b4)2 = (b×b×b×b)2 34×(32)3 = (b×b×b×b)×(b×b×b×b) = b8 (b4)2b3 or (b4)2 = b4×2 = b8 1 (22)3 5 (a2)2 9 (52)2 13 (53)3 17 (103)2 21 22×(22)3 25 d3(d2)2 29 (52)2×5

2 (23)3 6 (b3)4 10 (s2)5 14 (102)3 18 (h4)3 22 22×(23)3 26 b4(b3)4 30 s2(s2)5

Index Law 3 (am)n = am×n

= 34×36 = 310 = b8×b3 = b11

3 (32)3 7 (t2)4 11 (m2)3 15 (10)3 19 (d2)4 23 (32)3×33 27 (t2)4t5 31 m(m2)3

4 (42)3 8 (n2)5 12 (34)2 16 (g2)5 20 (25)3 24 44×(42)3 28 n3(n2)5 32 3×(34)2

Index Law 4 Zero Index: p3÷p3 = 1 or p3÷p3 = p3−3 = p0 Which must be = 1

or

Exercise 1.9 Simplify each of the following: 30 = 1

h0 = 1

1 20 5 a0 9 5×20 13 7w0 17 (50)2×5

2 50 6 30 10 3a0 14 3e0 18 s2(s0)5

Zero Index

Zero Index:

a0 = 1

p0 = 1 Try 50 on your calculator. Is your answer 1?

3×50 = 3×1 = 3 3 b0 7 100 11 6×40 15 9×30 19 m(m2)0

5b0 = 5×1 = 5 4 k0 8 d0 12 2×10 16 8×20 20 3×(30)2

Chapter 1 Indices 1

5

Index Law 5 What happens when the index is negative?

or a ×a 1 = a ×a ×a ×a ×a a ×a ×a

a2÷a5 =

Negative Index

a2÷a5

=

a2−5

a−m =

= a−3

1 am

Exercise 1.10 Write each of the following using a negative index: 1

= 10−3

103

1 5 9

1 1 1 4

6



1 x

= b−5

2



102 x7

1 b5

1 25 1 x

3



7



1

1 a4

= 23 = 2−3

{23=2×2×2=8}

1

4



1

104 1

8



106 1

10 9



1

1 8

= x−1

34 1

11 27

12 16

Find the value of each the following: 1

1

1

2−3 = 2 × 2 × 2 = 8 {also = 0.125} 13 2−2 17 10−2 21 5−2

1

10−4 = 10 ×10 ×10 ×10 = 10000 {also = 0.0001}

14 4−1 18 10−3 22 2−5

15 2−4 19 10−4 23 0.4−2

16 10−1 20 10−5 24 1.5−2

Copy each of the following tables and use a calculator to complete them: 25 Power 24 23 22 21 20 2−1 2−2 Value

6

4

2

2−3

26

Power Value

54

53

52 25

51 5

50

5−1

5−2

5−3

27

Power Value

104

103

102 100

101 10

100

10−1

10−2

10−3

Index Law 1 am×an = am+n This only works if the bases are the same.

Exercise 1.11 Use the Index Laws to simplify each of the following: 22×2−5 = 22+ˉ5 {am×an = am+n}

= 2−3

1 23×2−6 5 x7×x−3 9 48×4−6

{2 + ˉ5 = ˉ3}

5x−4×3x3 = 15xˉ4+3 {am×an = am+n}

2 52×5−6 6 2x3×3x−4 10 a5×a−6×a4

= 15x−1 {ˉ4 + 3 = ˉ1} 3 35×3−2 4 104×10−5 7 5x5×x−6 8 4x3×5x−5 11 3x3×2x−2×4x−4 12 103×10−3 4+ˉ5 = ˉ1 ˉ3+ˉ2 = ˉ5 5 −ˉ3 = 8 ˉ2−ˉ4 = 2

Index Law 2 am÷an = am−n

As a warmup, cover the answers. Can you get the correct answers?

52÷2−3 = 22−ˉ3 {am÷an = am−n}

= 25

13 33÷3−2 17 x3÷x−3 21 4x5÷2x−1

8x−5÷2x3 = 4xˉ5ˉ3 {am÷an = am−n}

{2 − ˉ3 = 2+3 = 5} 14 25÷2−2 18 x−3÷x5 22 12b4÷4b−2

= 4x−8

{ˉ5 − 3 = ˉ8}

15 5−3÷52 19 y−3÷y−4 23 24×23÷2−2

16 10−4÷10−2 20 a3÷a−1 24 103÷10−3 2×5 = 10 4×ˉ3 = ˉ12 ˉ3×2 = ˉ6 ˉ5×ˉ3 = 15

Index Law 3 (am)n = am×n

As a warmup, cover the answers. Can you get the correct answers?

(3−2)4 = 3ˉ2×4 { (am)n = am×n}

= 3−8

25 (2−2)3 29 (x2)3 33 x−3×(x−1)−3

{ˉ2×4 = ˉ8}

(x2)3 = x2×3 { (am)n = am×n}

= x6

{2×3 = 6}

26 (3−3)2 27 (22)−4 30 (n−3)2 31 (a4)−3 34 (3−3)2 ×(3−2)−2 35 a4×a−3×(a4)−3

28 (5−2)−2 32 (y−1)−5 36 (10−1)−2

Chapter 1 Indices 1

7

Summary 24×22 = 24+2

Index Law 1 am×an = am+n

=1

Index Law 5 a−m =

1



(73)2 = 73×2

30

= 2a (x3)2 = x3×2

= 76



Index Law 4 a0

6a3÷3a2 = 2a3−2

= 5−3



= x6



52÷55 = 52−5

Index Law 3 (am)n = am×n

= 26



Index Law 2 am÷an = am−n

3x4×2x2 = 6x4+2

= x6

x0

=1

=1



10−3 =

1

x−5 =

3

10

1 x5

am

The more problems you work, the better you become, the more opportunities.

Exercise 1.12 Use the Index Laws to simplify each of the following: 1 34×33

2 x2×x3

5 a2×a2

6 b3×b5

7 z5×d2

8 104×103

9 2−3×21

10 1.52×1.54

11 53÷52

12 105÷102

13 d4÷d2

14 22÷24

15 a2÷a6

16 e5÷x6

17 1.27÷1.22

18 z3÷z8

19 4−5÷44

20 103÷10−2

21 (32)2

22 (x3)2

23 (52)3

24 (x2)5

25 (a2)5

26 (s−2)5

27 (m2)−3

28 (34)2

29 (x−2)−4

30 (102)3

31 4a2×3a3

32 2b5×5b2

33 45÷42

34 10z3÷2z5

35 105×102×104

36 105×102×10−4

39 x4×6x−3÷2x2

40 (x−4)2×9x3÷6x2

37 106×10−2×10−4 38 3x4×2x3÷x2

8

3 72×75

4 x4×y3

The bases aren't the same. The answer is: x4×y3

Mental Computation You need to be a good mental

Exercise 1.13 athlete because many everyday 1 Spell power. problems are solved mentally. 2 5 − 7 3 3 − ˉ4 4 102×103 5 z3÷z7 10% of $6 = $0.60 6 (23)2 7 Increase $6 by 10% 8 I buy a loaf of bread for $3.45 and a biro for 40c, what is the total? 9 If I paid $50 deposit and 10 payments of $20. How much did I pay? 10 I spend $70 per week on groceries, roughly how much per year? Roughly 50 weeks $70×50 = $70÷2×100 = $35×100 = $3500

Exercise 1.14 1 Spell simplify. 2 ˉ3 + 5 1 googol = 10100 = 10 ­000 ­000 ­000 ­000 ­000 ­000 ­000 3 1 − ˉ2 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 4 105×102 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 ­000 5 7 5 x ÷x ­000 ­000 6 (42)−3 7 Increase $8 by 10% 8 I buy a loaf of bread for $3.35 and 2L milk for $3.10, what is the total? 9 If I paid $100 deposit and 10 payments of $15. How much did I pay? 10 I spend $120 per week on groceries, roughly how much per year?

Q: Divide 14 sugar cubes into 3 cups of coffee so Exercise 1.15 that each cup has an odd number of sugar cubes. 1 Spell negative. A: 1,1,12 2 ˉ4 + ˉ5 12 isn't odd! 3 5 − ˉ1 It's an odd number of cubes to put in a cup of 4 10−2×105 coffee. 5 a6÷a4 6 (33)3 7 Increase $9 by 10% 8 I buy a loaf of bread for $3.25 and a book for $9.50, what is the total? 9 If I paid $100 deposit and 10 payments of $35. How much did I pay? 10 I spend $80 per week on unleaded petrol, roughly how much per year?

Chapter 1 Indices 1

9

NAPLAN Questions Exercise 1.16 1 Calculate each of the following: b) 2.35×104 a) 2.35×102

c) 2.35×107

2 5.163×2.734 is closest to: a) 800 b) 8 000

c) 80 000 (ˉ2)2 = ˉ2×ˉ2 =4 (ˉ2)3 = ˉ2×ˉ2×ˉ2 = ˉ8 (ˉ2)4 = ˉ2×ˉ2×ˉ2×ˉ2 = 16

3

What is the value of 6a2 when a = ˉ2?

4

What is the value of 2x2 + 3x − 2 when x = ˉ1?

5

What is the value of a2 + b2 when a = 3 and b = ˉ1?

6

Is y = 3x3 a correct rule for y in terms of x?

7

Is y = 2x2 − x a correct rule for y in terms of x?

8

The volume of a cone is given by the formula: V = 3 , where r is the radius and h is the height. What is the volume of a cone with r = 8 cm and h = 12 cm?

x y

0 0

1 3

2 24

3 81

x y

0 0

1 1

2 6

3 15

πr 2 h

12 cm

8 cm

9

4 200 000 is the same as: a) 4.2×105 b) 4.2×106

c) 4.2×107

10 What is the value of: 3×103 + 7×102 + 6×101? 11 Calculate each of the following: a) 23 − 22 b) 33 − 32

c) 43 − 42

12 Which is the same as 32×32 a) 3×3×3 b) 3×3×3×3

c) 3×3×3×3×3

13 Which is the same as 23×42 a) 2×3×4×2 b) 2×2×2×4×2

c) 2×2×2×2×2×2×2

14 152 is between: a) 150 and 200

b) 200 and 250

c) 250 and 300

15 Solve for b in: a) 2b = 16

b) 2b=64

c) 3b = 81

16 What is the difference between 106 and 105?

10

Competition Questions Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 1.17 1 Find the value of each of the following: a) (0.1)2 b) (0.01)2 3 d) (0.1) e) (0.01)3 2 g) (ˉ1) h) (ˉ1)3 j) (ˉ1)5 k) (ˉ1)6 2 3 m) 10 + 10 +10 n) 2 + 22 + 23 + 24 2 2 p) 2 −1 q) 32−22

c) (0.001)2 f) (0.001)3 i) (ˉ1)4 l) (ˉ1)13 o) 10×102×103 r) 42−32

How many digits in 10.16? 10.12 = 102.01 3 + 2= 5 digits 10.13 = 1030.301 4 + 3= 7 digits 10.14 5 + 4= 9 digits

10.16 would have 7+6 = 13 digits

2

How many digits in? a) 10.17

3

The following sequence is formed by squaring the previous number and subtracting 5. What is the fifth term? 3, 4, 11, .....

4

Calculate each of the following:



a)



26 × 23

2 2 × 27



b) 10.137

b)

34 ×33

32 ×34

c) 10.311



c)

45 × 43 43 × 44

What is the last digit in 2315?

31 = 3 32 = 9 33 = 27 34 = 81 45 = 243

Last digit pattern is: 3, 9, 7, 1, 3, 9, 7, 1, .... repeating in blocks of 4 The index, 15, is one from the fourth repeat The last digit in 2315 is 7

5 6 7 8

What is the last digit in 332? What is the last digit in 4347? What is the last digit in 39388? What is the last digit in 9471?

9

Given that a and b can be any positive integer between 2 and 5, what is the largest possible value of (2a − b)(b−a)?



Chapter 1 Indices 1

11

Investigations Investigation 1.1

Maths Joke?

An infinite number of Year 9 Maths students go to the Tuckshop. The first goes up and asks, "I'll have a litre of milk, please." Each student, in sequence, says, "and I'll have half of what they had." The tuckshop person says, "You are all idiots," and gives them two litres of milk.

To understand the joke, calculate how much milk is needed: 1

1

1

1

∞ means infinity, an

extremely large number.



= 1 + 2 + 2 × 2 + 2 × 2 × 2 + 2 × 2 × 2 × 2 + .......

or

=1 + 2−1 + 2−2 + 2−3 + 2−4 + 2−5 + ....................... + 2−∞ What is the answer?

Investigation 1.2

Large numbers?

There are many legends in which the reward for a great deed is something like: $1 is paid on the first day of the month, and then doubled for each sucessive day of a 31 day month. $1

$2

$4

$8

What is the total payment for the month?

If a $1 coin is 9 grams, and the daily reward must be carried without assistance, out of the bank, what is the largest amount that will be paid?

12

A Couple of Puzzles Exercise 1.18 1 Give an estimate of: 31.3 x 4.87 2

Make 8.9 appear on the display of a calculator without using the 8 key or the 9 key.

3

A club's middle batsman named Chuck squared his number of runs just for luck by subtracting his score and forty-two more the final result was a duck

4

What is 2 of 2 of 2 of a pie?

1

1

How many runs did Chuck score?

1

A Game Knots is played with naughts and crosses on a 4x4 square. The loser is the person who can’t make a move. O

1 Take it in turns to place a naught or cross in one of the 16 cells.

X

O

X

O

O

2 A naught cannot be placed above, below, or beside another naught (diagonally is OK). Similarly for crosses.

X O

X

3 It is X’s turn. Can you find a place to put an X? O has lost the game.

A Sweet Trick 1 2 3

With a rope or string make the first knot as shown. Then add the second knot as shown in the middle diagram. Then finally add the final loops as shown.

Get your audience to pull each end of the rope.

Can you guess what happens?

Chapter 1 Indices 1

13

Technology Technology 1.1 Use a spreadsheet to help with Investigation 1.1. 1

1

1

1



= 1 + 2 + 2 × 2 + 2 × 2 × 2 + 2 × 2 × 2 × 2 + ................

or

=1 + 2−1 + 2−2 + 2−3 + 2−4 + 2−5 + ....................... + 2−∞

1 2 3 4 5 6 7 8 9 10

Amount Milk 1 0.5 0.25 0.125 0.0625

Running Total 1 1.5 1.75 1.875

= b2/2 = sum($b$2:b3)

1.3E−8 means move the decimal point 8 places to the left. =0.000 000 013 (or make the cell width wider)

Graph the running total.

Technology 1.2 Use a spreadsheet to help with Investigation 1.2. Day of Month 1 2 3 4 5 6 7 8 9 10

14

Reward 1 2 4 8 16 32

Daily Total 1 3 7 15 31

= 2*b2 = sum($b$2:b3)

1.3E+8 means move the decimal point 8 places to the right. =130 000 000 (or make the cell width wider)

Chapter Review 1 Exercise 1.19 1 Write each of the following in index form: 3×3×3×3 = 34

ˉa×ˉa×ˉa= (ˉa)3

a) 1×1×1×1 d) g×g×g×g×g×g×g 2

= 2−3

{2 + ˉ5 = ˉ3} b) 62×6−7 f) 4a2×3a−4 j) x3×x−6×x4

= 25

= 15x−1 {ˉ4 + 3 = ˉ1}



c) 24×2−2 d) 106×10−5 g) 2x5×x−6 h) 2x2×3x−5 k) 4x3×2x−2×x−5 l) 103×10−2 8x−5÷2x3 = 4xˉ5ˉ3 {am÷an = am−n} = 4x−8

{2 − ˉ3 = 2+3 = 5}

m) 23÷2−4 q) n3÷n−3 u) 6x6÷2x−1

n) 75÷7−4 r) x−4÷x5 v) 8x4÷4x−3

o) 3−3÷32 s) p−3÷p−4 w) 34×34÷3−2

{ˉ5 − 3 = ˉ8} p) 10−5÷10−2 t) a3÷a−2 x) 104÷10−4

Use the Index Laws to simplify each of the following: (3−2)4 = 3ˉ2×4 { (am)n = am×n}



c) 10×10×10×10×10×10 f) ˉ5×ˉ5×ˉ5×ˉ5×ˉ5×ˉ5

5x−4×3x3 = 15xˉ4+3 {am×an = am+n}

52÷2−3 = 22−ˉ3 {am÷an = am−n}

3

Base

Use the Index Laws to simplify each of the following:

a) 43×4−2 e) x6×x−3 i) 23×2−5



d3

b) 4×4×4×4 e) 1.7×1.7×1.7×1.7

22×2−5 = 22+ˉ5 {am×an = am+n}

Index

= 3−8

{ˉ2×4 = ˉ8}

(x−2)−3 = xˉ2×ˉ3 { (am)n = am×n}

= 26

{ˉ2×ˉ3 = 6}

a) (3−2)3

b) (2−2)−3

c) (y2)3

d) (10−1)−3

e) 64×62

f) x2×x5

g) 107×103

h) 2−5×21

i) 1.23×1.25

j) 45÷42

k) 106÷102

l) 8.13÷8.12

m) 3−5÷33

n) 109÷10−2

o) 4x2×2x4

p) 2x5×5x3

q) 106×102×10−4

r) 105×10−2×10−4

s) 5x2×2x3÷x2

t) 2x5×6x−3÷2x2

u) x−3×(x−1)−3

v) (x−4)2×4x3÷2x2

106 = 1 000 000 (generally named 1 million).

Chapter 1 Indices 1

15

Chapter Review 2 Exercise 1.20 1 Write each of the following in index form: 3×3×3×3 = 34

ˉa×ˉa×ˉa= (ˉa)3

a) 1×1×1×1×1 d) x×x×x×x×x×x 2

= 2−3

{2 + ˉ5 = ˉ3} b) d4×d−3 f) 3c2×2c−6 j) x−7×x3×x4

= 25

= 15x−1 {ˉ4 + 3 = ˉ1}



c) 25×2−2 d) 107×10−4 g) 5x3×x−4 h) 7x6×2x−5 k) 6x4×2x−3×x−5 l) 106×10−2 8x−5÷2x3 = 4xˉ5ˉ3 {am÷an = am−n} = 4x−8

{2 − ˉ3 = 2+3 = 5}

m) 33÷3−6 q) m3÷m−4 u) 10x6÷5x−2

n) w5÷w−8 r) x5÷x5 v) 6x4÷3x−5

o) 2−4÷22 s) 7−3÷7−4 w) 24×24÷2−5

p) 10−8÷10−3 t) x3÷x−2 x) 109÷10−4

= 3−8

{ˉ2×4 = ˉ8}

(x−2)−3 = xˉ2×ˉ3 { (am)n = am×n}

= 26

{ˉ2×ˉ3 = 6}

a) (2−3)3

b) (8−2)−3

c) (x4)2

d) (10−2)−3

e) 34×32

f) x7×x5

g) 105×106

h) 4−5×44

i) 6.73×6.79

j) 55÷53

k) 106÷105

l) 2.13÷2.15

m) 3−6÷33

n) 108÷10−2

o) 2x2×6x5

p) 3x7×5x3

q) 104×102×10−5

r) 107×10−3×10−5

s) 7x3×2x3÷x4

t) 5x4×6x−3÷2x2

u) (3−3)2 ÷(3−2)−2

v) a4×a−3÷(a5)−3

109 = 1 000 000 000 (generally named 1 billion).

16

{ˉ5 − 3 = ˉ8}

Use the Index Laws to simplify each of the following: (3−2)4 = 3ˉ2×4 { (am)n = am×n}



c) 10×10×10×10×10 f) ˉ3×ˉ3×ˉ3×ˉ3×ˉ3×ˉ3

5x−4×3x3 = 15xˉ4+3 {am×an = am+n}

52÷2−3 = 22−ˉ3 {am÷an = am−n}

3

Base

Use the Index Laws to simplify each of the following:

a) 53×5−4 e) x5×x−2 i) 13×1−9



d3

b) 2×2×2×2 e) 6.9×6.9×6.9×6.9

22×2−5 = 22+ˉ5 {am×an = am+n}

Index

Number and Algebra → Patterns and algebra  Apply the distributive law to the expansion of algebraic expressions, including binomials, and collect like terms where appropriate.  Understand that the distributive law can be applied to algebraic expressions as well as numbers, and understanding the inverse relationship between expansion and factorisation.  Extend and apply the index laws to variables, using positive integral indices.

A TASK Can you solve: x2 + 2x = 15?

Quadratic equations play a very important part in solving thousands of problems in our modern society. Research quadratic equations. The quadratic equation : x2 − 5x = ˉ6 has two solutions: x= 2 and x = 3. 22 −5×2 = ˉ6 32 −5×3 = ˉ6

A LITTLE BIT OF HISTORY Algebra has its origins in the work of mathematicians in ancient Babylonia.

The clay tablets were written on with a blunt reed leaving wedge shaped indents. Cuneiform tablet means the writing is wedge shaped.

There are many Babylonian maths tablets that have survived 2000 years. The Babylonian solution to the quadratic equation: x 2 + bx = c

was: x =

−b + 2

 b 2   + c  2 

17

Algebra Warmup Algebra is fundamental to solving millions of real world problems.

No big deal. Looks pretty simple to me.

Exercise 2.1 Simplify the following expressions: 3x + 2x = 5x

1 4 7 10 13 16 19

2x + 4x 7x − 2x 19y + 8y 10x − 3x 8a + 5a 5x + x + 3x 9x + 2x − 3x − x

5y − 7y = −2y

9b + 5b − b = 13b

2 5 8 11 14 17 20

3 6 9 12 15 18

5a − 3a 12c + 7c 4w − 6w 5b − 10b 7x − 9x 8h − 3h − 2h 8a + 2a − 4a − 5a

The key is to only join together the terms that are alike.

5y2 + 4y + y − 2y2 = 3y2 + 5y

7a + 5 − 4a = 3a + 5

21 3x + 4 + 2x 23 8 + 5x − 2x 25 8x + 5x + 7 27 6x + 3d −5x + d 29 8b2 − 4 + 5b2 + 9 31 8xy + 2xy + 6 − 5 33 3x + 4 + 2x − 6x

22 7 + 4b + 2b 24 4b − 4a + 2b 26 7a + 4b − 4a + 2b 28 9 + x + 3x − 7 30 7x3 + 5 − 5x3 + 3 32 6as2 + 5d5 − 2as2 + 2d5 34 −14x + 6y + 9x − 8y + y Multiply the numbers. Multiply the letters.

Exercise 2.2 Simplify the following expressions: 4 × 2x = 4 × 2 × x = 8x

1 4 7 10

3 × 5x 4 × 3x 3x × 9 5h × 2b m

13 2 × 4n

18

3c + 6c 3z − 5z 2x + 6x d + 3d 3m + 6m ˉ6x + 2x + 3x

3d × 5e = 3 × d × 5 × e = 15de

2 5 8 11

2 × 7a 3 × 5b 2f × 4n 8t × 3d p

14 10r × 2

10b ×

3 6 9 12

1 5

= 10 × = 2b

3 × 6m p×4 7x × 2y g × 7k k

15 8m × 4

1 ×b 5

Index Law 1

Multiply the numbers. Multiply the letters.

am×an

=

5mn × ˉ2m2n

4a × 3a = 4 × 3 × a × a = 12a2

16 4x × 3x 18 5a × 3a 20 5x × −2x 23 −4a2 × −3a 26 8s3 × 2s2 29 5mn × −3m2n 32 −3p2d × −2pd

Exercise 2.3 Simplify the following expressions: 8x 2

=



= 4x

1 4 7 10 13

= 5 × ˉ2 × m × m2 × n × n = ˉ10m3n2

17 3d × 4d m x m2 = m x m x m = m3 19 7d × 3d 21 2x × −3x 22 −3x × 4x 24 6p × −2p3 25 −9w × 3w2 27 4x2 × 5x × 2x 28 3e × e2 × 2d 30 7pn × −4p2n 31 4ab × −6a2b 33 −4h2 × −4h 34 −4a2b2c × −5a2bc 9a ÷ 6a and

8x ÷ 2

+ times − = − − times + = − − times − = +

am+n

9a ÷ 6a

=

=

2 5 8 11 14

8a ÷ 2 15x ÷ 5 24k ÷ 6 20x ÷ 15x 12ay ÷ 4a

9a 9 3 and and are the same thing. 6a 6 2

9a 6a

18ay ÷ 4a

3 2

=

12x ÷ 3 14y ÷ 7 30d ÷ 10 14g ÷ 4g 14de ÷ 4e

Divide the numbers. Divide the letters

−6x5

÷ 4x2y =

=

16 19 22 25

−8x

−6 x 5 4x 2 y

−3x 3 2y

÷ 4 −4g ÷ 2g −12a7y ÷ −4a3 −16b8c2 ÷ 24b6

3 6 9 12 15

=

18ay 4a 9y 2

6c ÷ 2 8n ÷ 4 9x ÷ 6 6y ÷ 4y 18dg ÷ 4d

+ divided by − = − − divided by + = − − divided by − = +

Index Law 2 am÷an

=

am−n

17 6p ÷ −3 20 −12a5 ÷ −4a2 23 14de4 ÷ −7e2 26 −28a5d3 ÷ −12d2

Calculators are very good at handling fractions: See Technology 2.1

18 −12y ÷ −2y 21 8v7 ÷ −4v4 24 −18dg ÷ 4d 27 −24d5w6z ÷ 36d3w3 Chapter 2 Algebra

19

Distributive Law The distributive law: a(b + c) = ab + ac

Each term inside the brackets: b and is multiplied by the term outside the brackets: a to give: ab + ac ie.,

a is distributed through the brackets.

Exercise 2.4 Expand each of the following: 4(a + 3) = 4a + 12 3(2b − 5) = 6b – 15 1 4 7 10

4(b + 3) 3(g + 1) 2(2z − 4) 4(3f − 7)

ˉ4(a + 3) = ˉ4a − 12 13 −2(a + 3) 16 −5(2m − 4) 19 −3(2c − 4)

2 5 8 11

t(2t + 3) 2d(3d − 4) −t(2t + 3) −p(3 + 2p)

3 6 9 12

5(c + 2) 6(h + 5) 5(2s − 4) 8(2a − 5)

14 −5(r + 2) 17 −3(y + 2) 20 −4(3e − 4)

5w(3w − 2m) = 15w2 – 10mw 22 25 28 31

Multiply each inside term by the outside term.

2(a + 7) 7(n + 4) 3(4d − 3) 6(5h − 6)

ˉ3(2b − 5) = ˉ6b + 15

5w x 3w = 15ww = 15w2

23 26 29 32

15 −2(c + 4) 18 −9(w + 6) 21 −5(4v + 3) + times + = + + times − = − − times + = − − times − = +

24 27 30 33

3z(4z + 5) 3n(2n − 4) −3z(4z − 2) −4e(3e − 2c)

6g(4g + 5) m(7m − 2) −6g(4g + 5) −4u(2b − 4u)

Exercise 2.5 Simplify each of the following by expanding and then collecting like terms: 1 3 5 7 9

20

8(2x + 3) + 5x + 7 = 16x + 24 + 5x + 7 = 21x + 31 2(x + 3) + 3x + 5 3(a − 4) + 9a + 13 5(x − 2) − 3x − 10 4(y − 8) + 4y − 10 t(2t + 3) + 5t2 + 6t



3(5a − 2) + 3a − 9 = 15a − 6 + 3a − 9 = 18a − 15 2 4 6 8 10

c

4(x − 3) + 2x − 1 6(2b + 6) − 10b + 3 2(5x − 2) + 5x + 4 2y(3y + 1) + 8y2 + 3y + 2 3z(4z + 5) + 15z2 + 10z

Distribute - to spread out, to cover everything.

The Distributive Law:

a(b + c) = ab + ac

Exercise 2.6 Simplify each of the following by expanding and then collecting like terms: ˉ5 × 3 = −5

–5(x



3(x + 2) + 2(x + 4)

+ 3) + 3(x − 1)



= 3x + 6 + 2x + 8

= –5x − 15 + 3x − 3



= 5x + 14

= –2x − 18

3 × −1 = −3

1

2(x + 3) + 3(x + 1)

2

−5(x

+ 2) + 2(x + 4)

3

2(c + 4) + 3(c + 3)

4

−4(d

+ 5) + 3(d + 1)

6 8 10 12 14

−2(x

5 7 9 11 13

5(h + 1) + 2(h + 3) 3(m + 4) + 2(m + 2) 5(w − 2) + 3(w + 4) 4(e − 4) + 3(e + 5) −t(2t + 3) + t(3t − 4)

+ 7) + 2(x − 2) 3(y − 2) + 2(y − 3) −3(c − 4) + 3(c + 2) −5(v + 3) + −2(v − 4) −3z(4z − 2) + −2(z − 4)

+ times + = + + times − = − − times + = − − times − = +

Multiply each inside term by the outside term.

The Distributive Law:

a(b + c) = ab + ac + times + = + + times − = − − times + = − − times − = +

–4

× −1 = 4



4(x + 3) − 3(x + 4)



= 4x + 12 − 3x − 12 =x −3 × 4 = −12

–4(x

− 1) − 2(x − 4)

= –4x + 4 − 2x + 8 = ˉ6x + 12

15 3(x + 4) − 2(x + 2)

16 −4(x − 1) − 2(x − 2)

17 2(x + 5) − 3(x + 4)

18 −5(y + 2) − 2(y + 6)

19 4(a + 1) − 6(a − 2)

20 −2(b + 2) − 2(b − 3)

21 −5(n − 1) − 3(n − 4)

22 −7(y − 1) − 5(y − 2)

−2 × −4 = 8

Chapter 2 Algebra

21

Distributive Law Multiply each inside term by the outside term.

The Distributive Law:

a(b + c) = ab + ac

Exercise 2.7 Simplify each of the following by expanding and then collecting like terms:

(x + 5)(x + 4)



(x + 3)2 = (x + 3)(x + 3)



= x(x + 4) + 5(x + 4)



= x(x + 3) + 3(x + 3)



= x2 + 4x + 5x + 20



= x2 + 3x + 3x + 9



= x2 + 9x + 20



= x2 + 6x + 9

1 3 5 7 9

(x + 1)(x + 2) (x + 3)(x + 1) (x + 2)(x + 4) (x + 2)2 (2x + 1)(x + 1)

2 4 6 8 10

(x + 2)(x + 1) (x + 1)(x + 4) (x + 1)2 (x + 3)2 (2x + 1)(x + 2)

+ times + = + + times − = − − times + = − − times − = +

Simplify each of the following by expanding and then collecting like terms:

22



(x + 5)(x – 3)



(x – 4)2 = (x – 4)(x – 4)



= x(x – 3) + 5(x – 3)



= x(x – 4) – 4(x – 4)



= x2 – 3x + 5x – 15



= x2 – 4x – 4x + 16



= x2 + 2x – 15



= x2 – 8x + 16

11 13 15 17 19 21 23 25

(x + 3)(x – 1) (x + 2)(x – 2) (x + 5)(x – 1) (x + 1)(x – 3) (x – 1)2 (x – 3)2 (2x + 3)(x – 1) (2x – 2)(x + 1)

12 14 16 18 20 22 24 26

(x + 4)(x – 1) (x + 4)(x – 4) (x + 2)(x – 3) (x + 2)(x – 3) (x – 2)2 (x – 5)2 (3x + 1)(2x – 2) (3x – 1)(2x – 2)

Factorisation

The inverse of distribution is called factorisation. In factorisation, the highest common factor is taken from each term.

The common term, a, is taken out and put at the front.

Factorisation:

ab + ac = a(b + c) Exercise 2.8 Factorise each of the following:

2x + 6



8a + 4

6x2 + 9x



= 2(x + 3)



= 4(2a + 1)



1 4 7 10 13 16

2a + 6 5x + 10 3p + 9 6a + 3 15x + 3 5p + 40

2 5 8 11 14 17

3 6 9 12 15 18

2b + 4 3m + 6 5d + 20 10u + 5 18g + 6 3n + 27

Algebra is an essential tool in thousands of careers and is fundamental to solving millions of problems.

= 3x(2x + 3) 2c + 10 4n + 8 7h + 35 9r + 3 21s + 7 35x + 5

A bit of factorisation preparation.

Exercise 2.9 Find the highest common factor of each of the following pairs of terms: 3a and 6a The factors of 3a are: 3, a The factors of 6a are: 2, 3, 6, a

4ef and 8fg The factors of 4ef are: 2, 4, e, f The factors of 6a are: 2, 4, 8, f, g

The highest common factor = 3a

The highest common factor = 4f

1 4 7 10 13 16

3x and 6x 4b and 6ab 3g and 12 6p and 14pq 6 and 12y 16g and 8

2 5 8 11 14 17

4a and 8ab 3xy and 9y 12e and 4 4p and 16 5xy and 15yz 16h and 64hij

3 6 9 12 15 18

6ab and 10a 5s and 10d 8ab and 12abc 8a and 2b 15ef and 27fg 14rt and 35t Chapter 2 Algebra

23

Factorisation

Factorising is the inverse of distributing. The highest common factor, a, is taken out and put at the front.

Factorisation:

ab + ac = a(b + c) Exercise 2.10 Factorise each of the following:

4x − 12



6ab − 8b

7t2 − 5t



= 4(x − 3)



= 2b(3a − 4)



1 4a − 12 4 3g − 6 7 10a − 4 10 6u − 8 13 10uv − 15u 16 16h2 − 12h

2 2b − 6 5 7w − 28 8 10s − 5 11 8pq − 6p 14 18b − 6 17 21d3 − 14d

3 5x − 15 6 3d − 6 9 15bc − 3c 12 12x − 8 15 14c2 − 21c 18 24p3 − 12p2 + times + = + + times − = − − times + = − − times − = +

Exercise 2.11 Factorise each of the following:

24

= t(7t − 5)



ˉ5x − 30



ˉ6x2 + 15x



ˉ20p3 − 12p2



= ˉ5(x + 6)



= ˉ3x(2x − 5)



= ˉ4p2(5p + 3)

1 3 5 7 10 13 16 19

−5z − 10 −3x − 12 −5d − 20 −9m − 18 −2d + 6 −4x2 + 12x −8b2c + 2b −6a3 + 15a

2 4 6 8 11 14 17 20

−2a − 10 −6q − 18 −4b − 16 −2n − 22 −3w + 27 −5g2 + 25g −6q2 + 3q −12t4 − 15t2

For these problems, take out the negative common factor. 9 12 15 18 21

−5b − 35 −6p + 36 −2e2 − 26e −15ao2 + 5o −6p5 − 36p3

Mental Computation

You need to be a good mental athlete because many everyday problems are solved mentally.

Exercise 2.12 1 Spell Distributive. 2 Simplify: 2a + 6a 3 Expand: 3(x − 2) 4 Factorise: 6x + 4 5 5 − 7 6 102×103 7 (23)2 8 Complete: 4×17 = 4(10 + 7) = 4×10 + 4×7 = 9 134 × 11 10 I pay $300 per week on rent, roughly how much per year? Roughly 50 weeks $300×50 = $300÷2×100 = $150×100 = $15 000

134×11 = 1474 Write the 1st and last numbers: 1.........4 Sum each consecutive pair: 1+3=4, 3+4=7 Insert these between the 1st and last: 1474

Exercise 2.13 1 Spell Factorise. 2 Simplify: 8x + 3x 3 Expand: 4(x − 3) Small opportunities are 4 Factorise: 9x + 6 often the beginning of great 5 ˉ2 − 4 enterprises - Demosthenes. 6 105÷103 7 (x2)5 8 Complete: 5×28 = 5(20 + 8) = 5×20 + 5×8 = 9 2416×11 10 I pay $360 per week on rent, roughly how much per year?

Exercise 2.14 1 Spell Algebra. In what month do people eat the least? 2 Simplify: 6y + 4y February – it’s the shortest month. 3 Expand: ˉ4(x − 2) 4 Factorise: 6x + 8 5 5 + ˉ9 6 104×102 7 (x3)3 8 Complete: 3×42 = 3(40 + 2) = 3×40 + 3×2 = 9 6251×11 10 I pay $480 per week on rent, roughly how much per year? Chapter 2 Algebra

25

NAPLAN Questions Exercise 2.15 The value of = =

5x when 3x − 4 5× 3 3× 3 − 4 15 =3 5

The value of 3x2 − 2x + 5 when x = −1? = 3×(ˉ1)2 − 2×ˉ1 + 5 = 3×1 + 2 + 5 = 10

x = 3?

5x

1

If x = 4, what is the value of 2x − 3 ?

2

y = 10 − 3x, what is the value of y when x = 2.5?

3

What is the value of 3x2 - 5x + 1 when x = −1?

4

A = 3d2 What is the value of A when d = 5?

5

M=

6

Given that P = 50b + 250, what is b when P = 600?

7

25ϕ = 30 What is the value of ϕ?

8

y = 3x + 2 y = 4x − 1 What value of x satisfies both of these equations?

9

y = 5x − 2 y = 3x + 6 What value of x satisfies both of these equations?

5a , 9b

what is the value of M when a = 1.8 and b = 0.2? One method is to substitute values for x until both expressions are the same.

10 A rule for a pattern is multiply by four and then add 3. The first three numbers of this pattern are: 7, 11, 15, ... What is the fifth number in this pattern? 11 A rule for a pattern is to add six and then divide by five. The first three numbers of this pattern are: 1.4, 1.6, 1.8, ... What is the fifth number in this pattern?

12 If a = 3, what is the value of 6a? 13 Expand: 2(5x + 1) 14 Expand: ˉ3(4a − 1) 15 What is the value of a2 + b2 when a = ˉ2 and b = 3? 16 What is the value of 5x2 when x = −2?

26

Competition Questions

Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 2.16 1 Evaluate each of the following: a) 1 + 2 × 3 − 4 b) 15 − 12 ÷ 3 c) 2 × 5 − 6 ÷ 2 d) (10 + 2) × 5 − 5 e) ((((1 − 2) − 3) − 4) − 5) f) 6 − (5 − (4 − (3 − (2 − 1))))

Order of Operations: 1 ( ) brackets first. 2 × and ÷ from left to right. 3 + and − from left to right.

2 Simplify each of the following: a) 23×22 b) 25÷23 c) 22÷23×24 d) 37÷39×32 3 Simplify each of the following: a) 5x + 3y − 3x + 2y b) 3x − 4y − 5x + y c) (2a + b) − (a − 3b) d) 4(x −2) − 3(x + 5) e) (x − 1) − (1 − x) f) 2x(3x − 1) + 5x2



27÷25

= 27−5 = 22 or 4

2(x − 1) − 3(x − 4) = 2x − 2 − 3x + 12 = ˉx + 10

4

What is the next term in the following sequence? 64, 26, 43, .....

5

What is the next term in the following sequence? 729, 36, 93, .....

6

If 2(x+2) = 16, what is the value of x?

7

If 4(2x−1) = 8, what is the value of x?

8

If x and y are positive numbers, which of the following is the largest? a) (x + y)2 b) x2 + y2 c) x2 + xy + y2

Stockbrokers buy and sell shares and bonds for clients.

• Relevant school subjects are Mathematics and English. • Courses usually involve a Universtity Bachelor degree with a major in commerce/finance.

Chapter 2 Algebra

27

Investigations Investigation 2.1

The Distributive law: a(b + c) = ab + ac

1

Write an algebraic expression for the area of the rectangle shown on the right.

2

Write an algebraic expression for the area of each of the two rectangles shown on the right and sum them together.

3

What do you notice?

b +c a

b

c

a

a

Investigation 2.2 The Distributive law: a(b + c) = ab + ac Use centicubes or counters to prove the distributive law for each of the following instances: 1

2(3 + 1)

=



2 lots of (3 + 1)

is the same as

2x3 + 2x1 2 lots of 3

and 2 lots of 1

+ 2

3(2 + 1)= 3x2 + 3x1

3

2(4 + 2) = 2x4 + 2x2

4

4(3 + 2) = 4x3 + 4x2

Investigation 2.3

(a + b)(a + b) = a2 +2ab + b2 (a + b)(a − b) = a2 − b2

Use one of the above methods to prove either:

28



(a + b)(a + b) = a2 +2ab + b2

or

(a + b)(a − b) = a2 − b2

a a

b

b

A Couple of Puzzles Exercise 2.17 1 Who am I? If you subtract me from 56 and then double the answer, the result is 26. 2 Put the four weights on the pans so that the balance balances.

4

2

1

1

A Game Guess 1 One player thinks of a number from 1 to 20 and writes it on a piece of paper.

11

2 The other player asks a series of questions to find out the number. The first player can only answer Yes or No. 1 Is the number even? 2 Is the number >10? 3 Is the number > 14? 4 Is the number = 11? 3 Switch players. The winner is the player asking the least number of questions.

NO YES NO YES

A Sweet Trick 1 Ask your audience to write down two numbers less than 20. 2 They add 1st to 2nd to make a 3rd. 3 They add 2nd to 3rd to make a 4th. 4 Repeat until there are ten numbers. 5 Ask them to total all ten numbers. Before they finish you call out the answer: 1474

6 13 19 32 51 83 134 217 351 568 How? Multiply the seventh number by 11 (134×11). Why?

Chapter 2 Algebra

29

Technology Technology 2.1 Simplifying Fractions Scientific calculators are excellent in working with fractions: 1

Simplify

15 35

15

a bc

35

=

3r7

2

Simplify

18 4

18

a bc

4

=

4r1r2



To change to a vulgar fraction: 2ndF

3

Use a scientific calculator to simplify the following ratios: a) 3 : 9 b) 9 : 12 c) 16 : 24 d) 2.1 : 3.5 e) 14.4 : 12.6 f) 256 : 1024

a bc

meaning

3 7

meaning 4 12

to give 9r2 ie

9 2

Technology 2.2 Expanding and Factorising Graphics calculators are capable of expanding and factorising: 1 Choose expand from the algebra menu. 2 Enter the algebraic expression: 3(4x − 5) to produce 12x − 15 1 Choose factor from the algebra menu. 2 Enter the algebraic expression: 12x − 15 to produce 3(4x − 5) Technology 2.3 The Distributive Law and Factorising There are a considerable number of resources about the Distributive Law and factorising on the Internet. Try some of them. Technology 2.4 Substitution Use a spreadsheet to check your answers to previous exercises: Use any substituting value other than 0

1 2 3

30

A Substituting value 2x + 6 2(x + 3)

B 3 12 12

Enter the first expression =2*B1 + 6 Enter the second expression =2*(B1 + 3)

Chapter Review 1 Exercise 2.18 1 Simplify the following expressions: a) 5x − 2x b) 7x + 3x d) 6 + 9x − 2x e) 4b − 4a + 3b g) 5 × 3x h) 3x × 5 − 2 − k) 5y × −2y3 j) 4x × 5x m) 20x ÷ 5 n) 21y ÷ 7 p) −12x ÷ 3 2

s) −10a5 ÷ −4a2

c) 3x − 5x f) 3rs2 + 5x5 − 2rs2 + 3x5 i) −3a × 2a l) 2x2 × 3x × 2x o) −8g ÷ 2g

q) −14y ÷ −2y

r) 18ab ÷ 4a

t) −24d5e6f ÷ 36d3e3 ˉ3(2y − 5) = −6y + 15

Expand each of the following:

a) d) g) j) m) 3

5y − 7y = −2y

3(x + 2) 2(3d − 4) −5(3t − 4) −x(2x + 3) −g(2 + 5g)

b) e) h) k) n)

c) f) i) l) o)

5(a + 4) 7(3h − 4) −2(y + 3) −3x(4x − 2) −4x(3x − 2y)

6(y + 7) 3(2x − 5) −8(p + 4) −6m(4m + 5) −4x(2y − 4x)

Simplify each of the following by expanding and then collecting like terms:

a) c) e) g) i) k) m)

2(x + 3) + 5(x + 4) 5(y − 2) + 2(y + 4) −p(2p + 1) + p(3p − 4) (x + 3)(x + 2) (x + 5)(x – 1) (x + 2)(x – 2) (2x – 2)(x + 1)

b) d) f) h) j) l) n)

4 Factorise each of the following: a) 5x + 15 b) 3y + 6 d) 18x + 3 e) 18e + 6 g) 4n − 8 h) 2m − 6 j) 16y − 4 k) 10x − 5 m) 6t − 10 n) 12x − 8 q) 21x3 − 14x p) 14f2 − 12f s) −5a − 15 t) −2b − 8 w) −4g2 + 24g v) −4x2 + 12x z) −12h5 − 15h2 y) −6x3 + 15x

−2(x

+ 5) + 4(x + 1) − 3) + 3(y + 2) −5b(4b − 2) + −2(b − 1) (x + 1)2 (x + 1)(x – 2) (x + 3)(x – 3) (3x – 1)(2x – 3) −4(y

12ab − 8a = 4a(3b − 2)

c) 4a + 16 f) 14d + 7 i) 5x − 15 l) 15ab − 3a o) 8uv − 6u r) 24p3 − 18p2 u) −2c + 6 x) −2x2 − 20x −9x5 − 12x2 = ˉ3x2(3x3 + 4)

Chapter 2 Algebra

31

Chapter Review 2 Exercise 2.19 1 Simplify the following expressions: a) 9x − 2x b) 5x + 4x d) 3 + 6x − 3x e) 9b − 3a + 2b g) 6 × 2x h) 8x × 2 − 2 − k) 4k × −2k3 j) 2x × 3x m) 24x ÷ 6 n) 28o ÷ 7 p) −16x ÷ 4 2

s) −14a6 ÷ −7a4

q) −12a ÷ −6a

r) 20mn ÷ 4m

t) −24a5e6t ÷ 20a4e3 ˉ3(2y − 5) = −6y + 15

2(x + 5) 6(2w − 4) −4(3v − 2) −a(4a + 3) −p(2 + 3p)

b) e) h) k) n)

c) f) i) l) o)

4(z + 3) 9(3c − 5) −6(r + 1) −5x(2x − 3) −4y(3x − 2y)

5(y + 3) 4(2x − 1) −7(u + 5) −6d(2d + 3) −4y(2y − 4x)

Simplify each of the following by expanding and then collecting like terms:

a) c) e) g) i) k) m)

3(x + 1) + 2(x + 4) 6(w − 3) + 2(w + 5) −b(2b + 5) + b(3b − 4) (x + 2)(x + 3) (x + 4)(x – 1) (x + 1)(x – 1) (2x – 1)(x + 1)

b) d) f) h) j) l) n)

4 Factorise each of the following: a) 3x + 15 b) 2a + 8 d) 15c + 3 e) 18d + 9 g) 3f − 9 h) 3g − 6 j) 20m − 4 k) 15x − 5 m) 12u − 10 n) 14v − 8 q) 18x4 − 12x p) 12r2 − 15r s) −5b − 20 t) −2d − 10 w) −6f2 + 24f v) −6x5 + 12x z) −18h7 − 15h2 y) −12x3 + 15x

32

c) 4x − 8x f) 6fg2 + 3z4 − 5fg2 + 3z4 i) −5a × 3a l) 3x2 × 4x × x o) −6h ÷ 2h

Expand each of the following:

a) d) g) j) m) 3

5y − 7y = −2y

−3(x

+ 2) + 4(x + 3) − 3) + 5(f + 2) −4g(4g − 5) + −3(g − 1) (x + 1)2 (x + 2)(x – 1) (x + 2)(x – 2) (3x – 1)(2x – 1) −2(f

12ab − 8a = 4a(3b − 2)

c) 6b + 12 f) 12e + 6 i) 5h − 15 l) 15bc − 5b o) 10st − 6s r) 12y4 − 18y2 u) −3c + 6 x) −5x2 − 20x −9x5 − 12x2 = ˉ3x2(3x3 + 4)

Measurement & Geometry  Using units of measurement  Calculate the areas of composite shapes.  Understand that partitioning composite shapes into rectangles and triangles is a strategy for solving problems involving perimeter and area.  Analyse nets of prisms and cylinders to establish formulas for surface area.  Calculate the surface area of cylinders and right prisms and solve related problems.  Become fluent with calculation of area and identify that area is used in the workplace and everyday life.

A TASK Reduce surface area NOW!

Allen's Rule suggests that animals from colder climates usually have shorter limbs and smaller ears. The reason being that a smaller surface area to volume ratio reduces heat loss. Research Allen's Rule • What is Allen's Rule. • Find examples that may support Allen's Rule. • Design an experiment to demonstrate Allen's Rule. • What advantages would a high surface area to volume ratio be to an animal or plant?

A LITTLE BIT OF HISTORY 1847 Bergmann's Rule suggests that large animals are found in colder climates and small animals are found in warmer climates. 1877 Allen's Rule suggests that animals with a smaller surface area to volume ratio are better able to survive colder climates. 1937 Hesse's Rule suggests that animals with a larger heart to body weight are found in colder climates compared to animals in warmer climates.

Surface area/volume ratio Tetrahedron

SAV= 7.2

Cube SAV=6

Sphere SAV=4.8

33

Area Warmup Square

Rectangle

s s

s s

h

l

l Area = l×b

Area = s2

Triangle

b b

Area = ½bh

b

Exercise 3.1 Calculate the area of each of the following shapes: 5.8m 5.8m Area

=s = (5.8m)2 = 33.64 m2

Area

1

= l×b = 3.7cm × 2.3cm = 8.51 cm2

2

6m

2.3cm

3.7m

9m

2.3cm

4

5

6 2.2km

12m

8

Area = ½bh = 0.5×4.7cm×6.2cm = 14.57 cm2

3

3.7m

7

4.7cm

3.7cm

2

36m

6.2cm

2.3cm

3.7km

5.3cm 2.9cm

A kitchen bench top is 1.4 m by 2.9 m. How many square metres of laminate is needed to cover the top of the bench? A rectangular paddock is 124 m by 111 m. What is the area of the paddock in square metres and hectares (1 hectare = 10 000m2)? A hectare is the area of a square 100 m by 100 m.

9

A triangular road sign has a base of 20 cm and a perpendicular height of 32 cm. What is the area of the road sign? 10 A bedroom is 3.3 m by 2.8 m. How many square metres of carpet is needed to cover the floor of the room? 11 A paddock, in the shape of a triangle, has a base of 648 m and a perpendicular height of 457 m. What is the area of the paddock in square metres and hectares? 12 The builder wants to put a 2 m wide concrete path around the outside of a 12 m square building. What is the area of the path?

34

Composite Shapes

Composite shapes can be squares, rectangles, and triangles composed together.

Exercise 3.2 Calculate the area of each of the following composite shapes: = 10-4=6

h = 9-7=2

4m

7m

7m

4m

13 m

10 m

Area

= rectangle + square = lb + s2 = 7×6 + 4×4 m2 = 42 + 16 m2 = 58 m2

1

Area

2

= triangle + rectangle = ½bh + lb = 0.5×13×2 + 13×7 m2 = 13 + 91 m2 = 104 m2

3

23 cm

9 cm

12 cm

5m

8m

9m

27 cm

5m

11 cm

6 cm 22 cm

12 m 14 cm

4

5

6

42 cm

29 cm 8 cm

9m

6m 14 m

7

68 cm

21 cm

63 cm

8

47 cm

9 cm

9

36 cm 55 cm

4.2 m 5.5 m

22 cm 14 cm

5 cm

Chapter 3 Area

35

Prisms

Prisms are solid, or hollow, objects with two identical ends and rectangular sides.

Triangular Prism

=

2 triangular ends

+

3 rectangular sides

Rectangular Prism

=

2 rectangular ends

+

4 rectangular sides

Pentagonal Prism

=

2 pentagonal ends

+

5 rectangular sides

Exercise 3.3 Copy and complete the following table: Prism Ends Sides Triangular prism 2 triangles 3 rectangles Rectangular prism 2 rectangles Pentagonal prism Hexagonal prism Heptagonal prism Octagonal prism

Total faces 5

Exercise 3.4 Draw a net for each of the following solids:

or

1 2 3

36

Surface Area The surface area of a solid is the total area of each face of the solid. Exercise 3.5 Find the surface area of each of the following prisms: Surface area = 2 ends + 2 sides + (top + bottom) 3 cm = 2×6×3 + 2×3×9 + 2×(6×9) cm2 9 cm = 36 + 54 + 108 cm2 6 cm Surface area = 198 cm2 1 2 5 cm

4 cm 8 cm

11 cm

7 cm

3 4 3.6 m 7.4 m 4.2 m

6 cm 1.9 cm 8.5 cm

7.7 cm

5 6 36 cm 4.1 m

7.8 m 3.6 m

5 cm



Surface area 5 cm 10 cm 6 cm 4 cm Surface area

= 2 ends

+

7 cm 3 sides

= 2×½×6×4 + 5×10 + 5×10 + 6×10 cm2 =

24

+

50 + 50

+ 60 cm2

= 184 cm2

7 8

10 cm

6 cm 16 cm

10 cm 18 cm 8 cm

12 cm

14 cm Chapter 3 Area

37

Cylinders

Prisms are solid, or hollow, objects with two identical circular ends and a tube side.

=

Circular prism

+

2 circular ends

1 tube

Circle Review Exercise 3.6 Calculate the circumference of each of the following circles (2 decimal places): C = 2πr C = πD = 2 × π × 1.5 = π × 5.4 1.5 m 5.4 cm = 9.42 m = 16.96 cm Using the calculator: C = 2 × π × 1.5

2

×

π

×

1.5

=

1 2 3 4 8.3

9 cm

mm

6.

1

cm

5 mm

Exercise 3.7 Calculate the area of each of the following circles (Correct to 2 decimal places): A = πr2 A = πr2 = π × 1.52 = π × (5.4÷2)2 1.5 m 5.4 cm 2 = 7.07 m = 22.90 cm2

Using the calculator: A = π × (5.4÷ 2)2

π

×

(

5.4

÷

2

)

x2

=

6. 1

5 mm

cm

1 2 3 4

38

9 cm

8.3

mm

Cylinder Surface Area

Surface area = 2 circles + rectangle. r

r

2πr

h

h

r Circular prism

Surface area

=

=

2 circular ends

2× π r 2

+

1 rectangle

+

2πr×h

Exercise 3.8 Find the surface area of each of the following cylinders: 4.3 m Surface area 6.9 m Surface area

= area of 2 circles + area of rectangle = 2×πr2 + 2πr×h = 2×π×4.32 + 2π×4.3×6.9 m2 = 116.18 + 186.42 m2 = 302.60 m2

1 2 5 cm 8 cm

9 cm 26 cm

3 4 9m 8m

5 6 1.6 m

2.1 m

7m 11 m

73 cm

42 cm

Chapter 3 Area

39

All Together

The surface area of a solid is the total area of each face of the solid.

Exercise 3.9 1 The four sides and the top of the 20 foot container are to be given two coats of paint. 20' Container Outside Dimensions Length 6.05 m Width 2.44 m Height 2.59 m a) b) c)

What is the surface area of the four sides and the top? How much paint is needed for two coats if 1 litre of paint will, on average, cover 15 m2. What is the cost of paint @ $63.00 for 4L?

2

Estimate the cost of painting the outside of the water storage tank. Assume that the painting (including labour, materials, scaffolding, etc) will cost $120/m2.

3

Calculate how many rolls of 70% shade cloth is needed to completely cover the greenhouse design.

8m

4m

1.1 m

70% shade cloth 1.83 m wide Roll (50m length)

$180 40

Mental Computation

You need to be a good mental athlete because many everyday problems are solved mentally.

Exercise 3.10 1 Spell Rectangular Prism. 2 What is the formula for the area of a triangle? 3 What is the area of a rectangle 4 m by 8 m? 4 What is the formula for the area of a circle? 5 Simplify: 3a − 5a 6 Expand: 3(x − 2) 7 Factorise: 6x + 4 8 102×103 9 107÷103 10 I buy a USB stick for $12.30 with a $20 note. How much change?

Why is 6 afraid of 7? Because 789

Exercise 3.11 1 Spell Surface Area. 2 What is the formula for the area of a rectangle? 3 What is the area of a triangle height = 4 m and base = 7 m? 4 What is the formula for the circumference of a circle? 5 Simplify: 4b × 3b 6 Expand: 2(3x − 1) 7 Factorise: 6x + 9 8 a4×a2 9 b2÷b6 10 I buy a DVD for $14.80 with a $20 note. How much change? Success is ninety-nine percent

failure - Soichiro Honda. Exercise 3.12 1 Spell Cylinder. 2 What is the formula for the area of a triangle? 3 What is the area of a rectangle 7 cm by 6 cm? 4 What is the formula for the area of a circle? 5 Simplify: 5a − 8a 6 Expand: 2(5x − 2) 7 Factorise: 4x + 10 8 107×103 9 (a2)3 10 I buy $35.60 of petrol with a $50 note. How much change?

Chapter 3 Area

41

NAPLAN Questions Exercise 3.13 1 Either make or draw a diagram of what each of the following nets will make. a) b) c)



d)

e)

2

What is the total surface area of the triangular prism?

f)

10 cm 14 cm 6 cm

h=12 m

16 cm πr 2 h

3

Find the volume of the cone. V = 3 r=3.2 m

6m

4

The area of the rectangle is 54 m2. What is the length of the rectangle?

5

What is the ratio of the area of the inner yellow square to the area of the larger green square?

6

What is the area of the floor plan?

7 What is the area of the trapezium?

2a a 4a

10m

4a a

42

3a

a

10m

10m

5m

Competition Questions

Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 3.14 1 The area of a square is 169 cm2. What is its perimeter? 2

Each floor of an office block has rectangular floors 32 m by 15 m. If the total office space is 3840 m2, how many floors in the building?

3

Find the area of each of the following shapes: 40 cm a) b)

10 cm square

10 cm

15 cm 25 cm

4

The overall yellow rectangle is 10 cm by 10 cm. What is the area of the triangle?

5

If the length of a rectangle has been doubled and the width tripled, what has happened to the area of the rectangle?

6

If the perpendicular height of a triangle has been tripled, what has happened to the area of the triangle?

7

How many 1 cm × 2 cm × 3 cm blocks can be cut from the block shown?

6 cm 5 cm 6 cm

8

The two rectangles overlap. What is the area of the overlap?

4a

a

3a 3a 4a

9 A circle is placed within a square as shown. What is the ratio of the area of the circle to the area of the square? a) 1:2 b) 2:π c) π:4 Chapter 3 Area

43

Investigations Investigation 3.1

Drawing 3D shapes

Rectangular prism a) Draw two rectangles

b) Join the vertices

Use a similar technique to draw: a) a triangular prism. b) a cylinder. c) a pentagonal prism. Investigation 3.2

3D shapes and Oblique Grids

Use oblique grid paper, isometric grid paper, and isometric dot paper to draw prisms. Which do you prefer?

These papers are available on the Internet.

Investigation 3.3 Rectangular prisms Bring a rectangular box from home.

1 2 3 4

Cut the rectangular faces from the box. Count the number of rectangular faces. Measure the rectangles and calculate their area. Calculate the surface area of the box.

Investigation 3.4 Newspaper revenue 1 Select a local newspaper. 2 Find the newspaper advertisement rates (or use the ad rate card shown). 3 Calculate the total revenue from the ads.

44

Advertising Rates Community Newspapers Full page = 16 modules $320 per module

Technology Technology 3.1 Surface Area Spreadsheets Make a spreadsheet to check your answers to the earlier exercises.

Enter the formula:

=2*b2*c2+2*c2*d2+2*b2*d2

Prism Rectangular

Length 6.01

Width 2.4

Height 2.7

Surface Area 74.262

Enter the formula:

=2*pi()*c2*b2+2*pi()*c2^2

Prism Cylinder

Length 4.92

Radius 1.2

Surface Area 46.14371

Technology 3.2 Prism Activities Search the Internet for some of the many prism activities. Use search phrases such as: a) 'Interactive geometry prisms' b) 'Polyhedra models' c) '3d objects nets' Technology 3.3

3D Sketching

"Mum, will you do my maths homework?" "No, it wouldn't be right." "Well, you could try."

a) Find 3D sketching software on the Internet and use the software to sketch/design in three dimensions. Use a search phrase such as '3d sketching'. b)

Use an Isometric drawing tool to draw 3D shapes. Use a search phrase such as 'Isometric drawing tool'.

c)

The use of a Vanishing Point makes drawings/sketches look more realsitic. Take part in an online tutorial about Vanishing Point drawings. Use a search phrase such as 'Vanishing point drawing'.

Vanishing Point

Chapter 3 Area

45

A Couple of Puzzles Exercise 3.15

1

1

= 0.01 , what is 10−3 1 If 10−2 = 2 = 100 10

2

Which arrangement of pipes will carry the most water? 2

6 3

A Game Double then nothing is a dice game in which the first person to 100 is the winner. 1

Roll a pair of dice. Your score is the sum of the two top faces of each die.

2

While it is your turn you can roll the dice as often as you like and you can keep totalling the score for each throw. However, if you throw a double then your turn is scored at zero and it is the next person's turn.

A Sweet Trick 1

Without looking, ask your audience to choose three different numbers from 1 to 9.

2 6 3

2

Have them write the three numbers in descending order.

632

3

Have them reverse the digits and find the difference.

632 − 236 = 396

4

Ask them to tell you either the first digit or the last digit.

3

The middle digit is always 9. The other two digits always sum to 9.

46

You tell them that the other two digits are 9 and 6.

Chapter Review 1 Exercise 3.16 1 Calculate the area of each of the following composite shapes: a) b) c) 5m

9m

5m

10 m

7m

4.7 m

15 m

13 m

6.1 m

2 Find the surface area of each of the following prisms: a) b) 20 cm

3 cm 9 cm

28 cm

6 cm

c)

16 cm

24 cm

d) 2.5 cm 7 cm 6 cm

3

3c

m 4c m

4 cm

The four sides and the top of the 40 foot container are to be given two coats of paint. 40' Container Outside Dimensions Length 12.19 m Width 2.44 m Height 2.59 m

a) b) c)

What is the surface area of the four sides and the top? How much paint is needed for two coats if 1 litre of paint will, on average, cover 15 m2. What is the cost of paint @ $74.00 for 4L?

Chapter 3 Area

47

Chapter Review 2 Exercise 3.17 1 Calculate the area of each of the following composite shapes: a) b) c) 6m

10 m

5m

13 m

9m

4.4 m

18 m

15 m

5.8 m

2 Find the surface area of each of the following prisms: a) b) 15 cm

4 cm 12 cm

26 cm

7 cm

c)

12 cm

18 cm

d) 4.8 cm 9 cm 9 cm

3

48

Estimate the cost of painting the outside of the water storage tank. Assume that the painting (including labour, materials, scaffolding, etc) will cost $110/m2.

4c

m 5c m

5 cm

Number and Algebra  Linear and non-linear relationships  Sketch linear graphs using the coordinates of two points.  Determine linear rules from suitable diagrams, tables of values and graphs and describe them both using words and algebra.  Sketch parabolas, hyperbolas, circles.

A catenary is the shape of a cat's tail?

A TASK Catenary means chain, and refers to the shape of a chain hanging between two supports. (e x + e−x ) A simplified equation of the catenary is: y= • • • • •

2

What is e? Use the formula to graph a catenary (Technology 4.3). Research practical applications of the catenary. Take photos of different kinds of catenaries. Publish your findings - classroom wall?

A LITTLE BIT OF HISTORY 1638 Galileo describes a hanging chain as similar in shape to a parabola. 1690 Huygens first uses the term catenaria. 1691 Leibniz, Huygens and Bernoulli derive the equation of the catenary after Bernoulli issued a challenge.

A spider's web has many catenaries.

49

Linear Rules

A linear rule increases by the same amount each step.

Exercise 4.1 For each of the following patterns: a) Count the number of matches or dots needed for each step, extend the pattern and summarise the pattern in a table. b) Write a rule for the pattern. c) Check that the rule is correct:

Table

Step Matches

1 4

2 7

Extra 3 matches for each step 3×step First step is 3+1 (= 4 matches). matches = 3×step+1

3 4 5 10 13 16

Rule: Matches = 3×step + 1 or m = 3s + 1 Check. When step = 5, Matches = 3×5+1 = 16 1

2

3

4

5

50



Linear Rules Exercise 4.2 Write a rule for each of the following tables of values: 1 ˉ6

x y

2 ˉ4

3 ˉ2

4 0

5 2

10 12

20 32

Extra 2 for each x → 2x

Check your answer: y = 2x − 8 y = 2×5 − 8 {Checking for x=5} y=2 

First step (ˉ6) is 2−8. → y = 2x − 8 1

x y

2

x y

3

x y

4

x y

1 4

2 6

3 8

4 5 10 12

10 22

20 42

1 4

2 7

3 4 5 10 13 16

10 31

20 61

1 2 3 ˉ5 ˉ3 ˉ1

4 1

5 3

10 13

20 33

1 2 3 ˉ7 ˉ4 ˉ1

4 2

5 5

10 20

20 50

1 2 3 14 11 8

x y

4 5

5 2

10 20 ˉ13 ˉ43 Check your answer: y = ˉ3x + 17 y = ˉ3×10 +17 {Check x=10} y = ˉ13 

Subtract 3 for each x → ˉ3x First step (14) is ˉ3+17. → y = ˉ3x + 17 5

x y

6

x y

7

x y

8

x y

A linear rule increases by the same amount each step.

1 15

2 13

3 11

4 9

5 7

10 ˉ3

1 5

2 2

3 ˉ1

4 ˉ4

5 ˉ7

10 20 ˉ22 ˉ52

1 3

2 ˉ1

3 ˉ5

4 ˉ9

5 10 20 ˉ13 ˉ33 ˉ73

1 ˉ7

2 ˉ5

3 ˉ3

4 ˉ1

5 1

10 11

20 ˉ23

20 31

Algebra is an essential tool in thousands of careers and is fundamental to solving millions of problems.

Chapter 4 Linear & Non-Linear Graphs

51

Linear Rules

The graph is linear because the points are in a line.

Exercise 4.3 Write a rule for each of the following graphs:

y

4



Write the coordinates in a table:

3

x y

2 1



-3 -2 -1

-1

1

2 3

x

-2

0 -3

1 -1

Increase of 2 each x → 2x

1 2 y

y 7

5

6

4

5

3

4

2

3

1 -3 -2 -1 -1

1

2 3

2

x

1 -3 -2 -1 -1

-2 -3

3 4

y

2

6

-3 -2 -1 -1

4 3

-2

2

-3

1 1

2 3

x

-4 -5 -6

-2 -3

Can you write the rule without having to complete a table?

52

1

2 3

x

y

1

5

-1

3 3

Step 1 (ˉ1) is 2−3. → y = 2x − 3

-3

-3 -2 -1

2 1

1

2 3

x

Sketching Linear Graphs A linear graph increases by the same amount each step.

Exercise 4.4 For each of the following: a) Copy and complete the following table. b) Draw a graph of the linear rule. x -2 y = 2x + 1

-1

0

1



-1 -1

0 1

Plot the points in the table.

y 4 3 2 1

1 3

-3 -2 -1 -1

2 5

1

x

2 3

-2 -3

The x first, then the y.

1 2 x -2 -1 0 1 2 y=x+2

3 4 x -2 -1 0 1 2 y = 2x + 5

5 6 x -2 -1 0 1 2 y = 2x − 3

7 8 x -2 -1 0 1 2 y = −x + 2

9 10 x -2 -1 0 1 2 y = 40x + 60

5

2×ˉ2 = two lots of ˉ2 = ˉ2 ˉ2 = ˉ4

When x = ˉ2, y = 2×ˉ2 + 1 = ˉ3 When x = ˉ1, y = 2×ˉ1 + 1 = ˉ1 When x = 0, y = 2×0 + 1 = 1 When x = 1, y = 2×1 + 1 = 3 When x = 2, y = 2×2 + 1 = 5 x -2 y = 2x + 1 -3

y = 2x + 1

2

x y=x+3

-2

-1

0

1

2

x y=x−1

-2

-1

0

1

2

x y = 3x + 1

-2

-1

0

1

2

x y = ˉ3x + 4

-2

-1

0

1

2

x y = 25x − 10

-2

-1

0

1

Chapter 4 Linear & Non-Linear Graphs

2

53

Linear Graphs

Linear Graphs are of the form:



y = mx + c

a) m is the increase each step. b) c is the value of y when x = 0 ie (0,c). c) The power of x and y is 1: y1 = mx1 + c

Exercise 4.5 Draw a quick sketch of each of the following: y = 2x + 1 y = mx + c Plot (0,1)

y

Across 1, up 2 (increase 2 each step)

5 4 3

m = 2 {increases by 2 each step} c = 1 {a point is (0,1)}

2 1 -3 -2 -1 -1

1

2 3

x

-2 -3

1 4 7

y = 2x + 2 y = 3x + 2 y = 2x − 3

2 5 8

y = ˉ2x + 3 y = mx + c

y = 3x + 1 y = 2x − 1 y = 4x + 5

3 6 9

y = 2x + 5 y = 5x + 2 y = x − 2

m=1

y

Plot (0,3)

5 4

Across 1, down 2 (decrease 2 each step)

3

m = ˉ2 {decreases by 2 each step} c = 3 {a point is (0,3)}

2 1 -3 -2 -1 -1

1

2 3

x

-2 -3

10 13 16

y = ˉ2x + 2 y = ˉ3x + 1 y = ˉ2x − 2

11 14 17

y = ˉ3x + 3 y = ˉ2x + 5 y = 3x + 2

12 15 18

3x + y = 5 y = ˉ3x + 5 {inverse of 3x is ˉ3x} y = mx + c

y = ˉ2x + 4 y = ˉ2x − 1 y = ˉx − 1 y 4 3

m = ˉ3 {decreases by 3 each step} c = 5 {a point is (0,5)}

2 1 -3 -2 -1 -1

54

2x + y = 1 x + y = 4 ˉ2x + y = ˉ2

20 23 26

ˉ2x + y = 1 x + y = 5 ˉ3x + y = ˉ1

Across 1, down 3 (decrease 3 each step)

5

Plot (0,5)

19 22 25

m = ˉ1

21 24 27

1

2 3

x

x+y=3 3x + y = 2 4x + 2y = 6

Non-Linear Graphs Parabolas - the shape of falling objects.

Exercise 4.6 For each of the following: a) Copy and complete the table. b) Plot the points. y = ˉx2 + 4 x y = ˉx2+4

ˉ2

ˉ1

0

1

2 5

When x = ˉ2, y = ˉ(ˉ2×ˉ2) + 4 = ˉ4 + 4 = 0 When x = ˉ1, y = ˉ(ˉ1×ˉ1) + 4 = ˉ1 + 4 = 3 When x = 0, y = ˉ(0×0) + 4 = 0 + 4 = 4 When x = 1, y = ˉ(1×1) + 4 = ˉ1 + 4 = 3 When x = 2, y = ˉ(2×2) + 4 = ˉ4 + 4 = 0 x y = ˉx2+4



ˉ2 0

ˉ1 3

0 4

1 3

y

4 3 2 1 -3 -2 -1 -1

2 0

x 1

2 3

-2 -3

b) Plot the points.

1 2 x -2 -1 0 1 2 y=x

2

3 4 x -2 -1 0 1 2 y=x +2 2

5 6 x -2 -1 0 1 2 y=x −1 2

7 8 x -2 -1 0 1 2 y = x(x + 1)

x y = x2 + 1

-2

-1

0

1

2

x y = x2 + 3

-2

-1

0

1

2

x y = x2 − 2

-2

-1

0

1

2

x -2 y = x(x − 1)

-1

0

1

2

Parabola:

y = ax2 + bx + c If a is positive: If a is negative:

Chapter 4 Linear & Non-Linear Graphs

55

Non-Linear Graphs

Circles - Points at a constant distance from the centre.

Exercise 4.7 For each of the following: a) Check that the table is correct. b) Plot the points. 2 x + y2 = 4

x2 + y2 = a2

x

ˉ2

ˉ1

0

1

2

x2 + y2 = 4

0

3 or ˉ 3

2 or ˉ2

3 or ˉ 3

0

y When x = 1, y = ˉ 3

2

x2 + y2 = (1)2 + (ˉ 3 )2

=1+3



=4

x -2



2 -2

ˉ 3 = ˉ1.7

Thus (1,ˉ 3 ) is a point on x2 + y2 = 4

1

x

ˉ2

ˉ1

0

1

2

1 or ˉ1

2 or ˉ2

5 or ˉ 5

2 or ˉ2

1 or ˉ1

x

ˉ3

ˉ2

0

2

3

x2 + y2 = 9

0

5 or ˉ 5

3 or ˉ3

5 or ˉ 5

0

x2 + y2 = 5 2

3

4

x

ˉ4

ˉ2

0

2

4

x2 + y2 = 16

0

12 or ˉ 12

4 or ˉ4

12 or ˉ 12

0

x

x2 − y2 = 4

ˉ4

ˉ2

2

4

6

12 or ˉ 12

0

0

12 or ˉ 12

32 or ˉ 32 6

Hyperbolas:

y

5 4

x2 − y2 = a2

x

3 2 1

The shadow of the tip of a stick traces out a hyperbola on the ground over the course of a day.

-4 -3 -2 -1

-1 -2 -3 -4 -5 -6

56

1

2 3

4

5

6

Mental Computation

The majority of everyday problems are solved mentally by adults.

Exercise 4.8 1 Spell Linear. 2 What is the linear rule for:

x y

0 -3

1 -1

2 1

3 3

3 Roughly sketch the rule: y = 2x + 3 4 What is the formula for the area of a triangle? 5 What is the area of a rectangle 4 m by 8 m? 6 Simplify: 3a − 7a 14×15 = 14×(10+5) 7 Expand: 2(x − 3) = 14×10 + 14×5 8 Factorise: 6x + 4 = 14×10 + 14÷2×10 9 102×103 = 140 + 70 10 14×15 = 210 Exercise 4.9 1 Spell Parabola. 2 What is the linear rule for:

x y

0 5

1 3

2 1

3 ˉ1

3 Roughly sketch the rule: y = 3x + 2 4 What is the formula for the area of a circle? 5 What is the area of a triangle height = 6 m and base = 7 m? 6 Simplify: 7m − 3m 7 Expand: 3(x − 2) 8 Factorise: 3x + 9 Student: Pi r squared. 9 105×102 Baker: No! Pies are round, cakes are square! 10 18×15 Exercise 4.10 1 Spell Hyperbola. 2 What is the linear rule for:

x y

0 -2

1 1

2 4

3 7

3 Roughly sketch the rule: y = 2x − 3 4 What is the formula for the circumference of a circle? 5 What is the area of a rectangle 5 m by 7 m? 6 Simplify: 6b − 9b 7 Expand: 5(x + 3) 8 Factorise: 5x + 10 9 104÷102 One may walk over the highest mountain one step at 10 22×15 a time - John Wanamaker.

Chapter 4 Linear & Non-Linear Graphs

57

NAPLAN Questions Exercise 4.11 1 What is the rule for the number of matches?

1

2

3

2

A rule is: y = 5 − 2x. What is the value of y when x = 2.25?

3

A rule is: y = 5 − 2x − x2. What is the value of y when x = ˉ2?

4

Write a linear model of the following plumber's fee: Hours (h) Fee ($F)

1 90

2 140

3 190

4 240 y 5 4 3 2 1

5

Write the linear rule represented by this graph.

-3 -2 -1 -1

1

2 3

x

-2 -3

6

What is the rule for the power cost?

140 120

Cost ($)

100 80 60 40 20 0

7

2

3 4

5

0 ˉ1

1 1

2 3

6

Power units

Which rule would produce the following table? a) y=x+1 b) y=2x−1 c) y=x2−3 x y

58

1

3 5

7

Don't do too much in your head. Pen and paper work will get better results.

8

A rule is: y = 2a2 − 4b. What is the value of y when a = 2 and b = ˉ2?

9

A rule is: y = 5x2 − 4. What is the value of y when x = ˉ3?

Competition Questions Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 4.12 1 An approximate formula for converting kilometres,

5k . How many miles in 32 kilometres? 8



k, to miles, m, is m =

2

Does the point (ˉ2,1) lie on the line: y = 2x + 5?

3

The point (100, 8) lies on the line 20x + y = 2008. Find another point that lies on this line?

4

What is the equation of each of the following lines? Each step increases by 2 and it cuts the y-axis at (0,ˉ3). m=2 c=ˉ3 y = 2x − 3 a) Each step increases by 2 and it cuts the y-axis at (0,1). b) Each step decreases by ˉ2 and it cuts the y-axis at (0,ˉ1).



Write a mathematical model of the following printing costs:

No. books 2 000 3 000 5 000 10 000

Let b = no. of thousands Each step increases by $1500: When b = 0, cost=$4000

Thus C = 1500b



Thus C = 1500b + 4000

5

Write a mathematical model of the following printing costs:

Pressure (atmos)

6a) The following graph shows the relationship between water depth, in metres, and pressure, in atmospheres. What is the rule?

No. books 3 000 5 000 7 000 10 000

Cost $7 000 $8 500 $11 500 $19 000

Cost $12 000 $15 000 $18 000 $22 500

6b) Which line represents 3y= ˉ6x + 9? A

4 3

B

2 1

60

0

50

-1

40 30

1

2

3

4

5

C

-2

20

-3

10 0 20 40 60 80 100

Metres

-4

D

Chapter 4 Linear & Non-Linear Graphs

59

Investigations Investigation 4.1 A falling rock? The distance a rock falls when dropped is given by the parabola: d = 4.9t2, where d is the distance in metres dropped and t is the time in seconds.

Investigate

Example: A stone is dropped into a well and a splash sound is heard 2.5 seconds later. Distance = 4.9t2 = 4.9×2.52 m = 30.6 m

Dropping stones. Investigation 4.2 The Ellipse The Earth moves around the Sun in an elliptical orbit. Use the method shown below to draw an ellipse.

Ellipse: Investigate the formula: x 2 y2 + =1 9 16

Investigation 4.3

The Parabola

The parabola describes free motion through the air. • The flight of a cricket ball. • Water from a hose.

The parabola has many practical applications. Investigate one of them. Civil Engineers design, build, and maintain bridges, dams, railways, buildings, airports etc. • Relevant school subjects are English, Mathematics, and Science. • Courses usually involve a Universtity engineering degree.

60

Technology Technology 4.1 Use a Graphics Calculator to plot the rules in Exercise 4.5. x -2 y = 2x + 1

-1

0

1

2

and enter the equation eg., 2x + 1

Press

Y=

Press

Graph

to see a plot of the equation

Press

Table

to see a table of the values

x -2 y = 2x + 1 -3

-1 -1

0 1

1 3

2

5

Technology 4.2 Use a Graphics Calculator, as above, to experiment with non-linear tables of data (See Exercises 4.6 and 4.7). Technology 4.3 Use a Spreadsheet to experiment with a Catenary. x y = 0.5(ex + e-x)

-3 10.1

-2 3.8

1 1.5

Enter the formula: = 0.5*(EXP(B2)+EXP(-B2))

0 1

1 1.5

2 3.8

3

10.1

A catenary is the shape of a hanging rope. 12 10

Use the Chart (Scatter) to plot the points:

8 6 4 2 0 -4

-3

-2

-1

0

1

2

3

4

Technology 4.4 Applets There are a very large number of "graph, function, plotter" applets on the Internet. There are also applets that let you experiment with parabolas, hyperbolas, ellipses. Chapter 4 Linear & Non-Linear Graphs

61

A Couple of Puzzles Exercise 4.13 1 Who am I? If you double me and then add twenty-five, the result is seventy-three. 2

If a∆b = a×b − 6, what is 4∆3?

3

A cube has been made by gluing twenty-seven smaller cubes together. How many faces have glue on them?

A Game Fox and Geese is a two player game. One person is the fox, and the other person looks after the 13 geese. Taking turns, the fox and a goose move along a line to the next intersection. The fox kills a goose by jumping over the goose onto a vacant intersection, in line, on the other side of the goose. The goose is removed from the game. The geese try to herd the fox into a corner where the fox cannot make a move. A goose cannot jump over the fox.

A Sweet Trick 1 Ask your audience write down any number and then enter the number into a calculator. 729 2

Add 17

729+17 = 746

3

Multiply by 64

746×64 = 47744

4

Multiply by 25

47744×25 = 1193600

5

Subtract 22176

1193600−22176 = 1171424

6

Divide by 1 600

1171424÷1600 = 732.14

7

Subtract the original number This will give the value of π every time. Make up a story - This is another way of working out the value of π?

62

732.14−729 = 3.14

They need to press '=' after each calculation.

Chapter Review 1 Exercise 4.14 1 Write a rule for the following pattern:

2



Write a rule for each of the following tables: a) x 1 2 3 4 5 10 20 y 4 6 8 10 12 22 42

A pessimist says the glass is half empty. An optimist says the glass is half full. An engineer says, “Why all the wasted space?”

b) 1 5

x y

2 2

3 ˉ1

4 ˉ4

5 ˉ7

10 20 ˉ22 ˉ52

3 Write a rule for each of the following graphs: a) b) y

2

5

1

4

-3 -2 -1 -1

3 2

1

2 3

x

-2

1 -3 -2 -1 -1

y

1

2 3

-3

x

-4 -5

-2

-6

-3

4 Copy and complete each of the following tables and draw a graph of the rule: a) b) x y = 2x − 3

-2

-1

0

1

2

c)

-2

-1

0

1

2

x y = x2 − 2

-2

-1

0

1

2

d)

x y = x2 − 1

-2

-1

5 Draw a sketch of: a) y = 2x + 2 6

x y = ˉ3x + 4

0

1

2

b) y = 2x − 1

c) y = ˉ2x + 4

Sketch x2 + y2 = 9. x

x +y =9 2

2

ˉ3

ˉ2

0

2

3

0

5 or ˉ 5

3 or ˉ3

5 or ˉ 5

0

Chapter 4 Linear & Non-Linear Graphs

63

Chapter Review 2 Exercise 4.15 1 Write a rule for the following pattern:

2



Write a rule for each of the following tables: a) x 1 2 3 4 5 10 20 y 5 8 11 14 17 32 62

If your attack is going well, you have walked into an ambush - Murphy's Laws of Combat.

b) 1 7

x y

2 3

3 ˉ1

4 ˉ5

5 ˉ9

10 20 ˉ29 ˉ69

3 Write a rule for each of the following graphs: a) b) y

y

7

6

6

5

5

4

4

3

3

2

2

1

1 -3 -2 -1 -1

1

-3 -2 -1

x

2 3

-1

1

x

2 3

-2 -3

4 Copy and complete each of the following tables and draw a graph of the rule: a) b) x y = 2x − 1

-2

-1

0

1

2

c) -2

-1

5 Draw a sketch of: a) y = 2x + 1

-1

0

1

2

0

1

2

x y = 2x2 − 3

-2

-1

0

1

2

b) y = 2x − 2

c) y = ˉ2x + 2

Sketch x2 + y2 = 16. x

x + y = 16 2

64

-2

d)

x y = x2 − 1

6

x y = ˉ3x + 2

2

ˉ4

ˉ2

0

2

4

0

12 or ˉ 12

4 or ˉ4

12 or ˉ 12

0

Chapter 1 Indices 1 2×2×2 =

2

3

Index Base

Index Law 2

Index Law 1

(ˉ2)2 = ˉ2×ˉ2 =4 (ˉ2)3 = ˉ2×ˉ2×ˉ2 = ˉ8 (ˉ2)4 = ˉ2×ˉ2×ˉ2×ˉ2 = 16

Index Law 3

am÷an = am−n

am×an = am+n

Zero Index

Negative Index

a0 = 1

(am)n = am×n

a−m =

1 am

Chapter 2 Algebra 1 7a + 5 − 4a = 3a + 5

3d × 5e =3×d×5×e = 15de

Distributive Law: a(b + c) = ab + ac 3(2b − 5) = 6b – 15

4(x − 1) − 2(x − 4) = –4x + 4 − 2x + 8 = ˉ6x + 12

s

Square

4x − 12 = 4(x − 3)

ˉ6x2 + 15x = ˉ3x(2x − 5)

Rectangle

s

b

s

b

Area = l×b

Triangle

l

l

s

Area = s×s = s2

18ay 4a 9y = 2

=

Factorisation: a(b + c) = ab + ac

–

Chapter 3 Area

18ay ÷ 4a

5mn×ˉ2m2n = 5×ˉ2×m×m2×n×n = ˉ10m3n2

Cylinder

r h

h Area = ½bh

Surface Area = 2πr2+2πrh

b

Chapter 4 Linear & Non-Linear Graphs Linear Graphs are of the form:



4

y = mx + c

y

3 2 1

a) m is the increase each step. b) c is the value of y when x = 0 ie (0,c).

-3 -2 -1

-1

1

2 3

x

-2 -3



Increase of 2 each x → 2x When x=0, y=ˉ3 → y = 2x − 3

65

Review 1 Exercise 5.1 Mental computation 1 Spell Cylinder. x y

2 What is the linear rule for:

0 -2

1 0

2 2

3 4

3 Roughly sketch the rule: y = 2x + 1 4 What is the formula for the area of a triangle? 5 What is the area of a rectangle 3 m by 4 m? 6 Simplify: 5a − 7a 14×15 = 14×(10+5) 7 Expand: 3(x − 2) = 14×10 + 14×5 8 Factorise: 6x + 4 = 14×10 + 14÷2×10 9 102×105 = 140 + 70 10 14×15 = 210 Exercise 5.2 1 Write each of the following in index form: 3×3×3×3 = 34

ˉa×ˉa×ˉa= (ˉa)3

a) 1×1×1×1 d) a×a×a×a×a 2

b) 4×4×4×4×4×4 e) 1.3×1.3×1.3×1.3

= 2−3

a) 34×3−2 e) x7×x−3

= 25

i) 23÷2−5 m) x−3÷x5 Index Law 1 am×an = am+n

66

c) 10×10×10×10×10 f) ˉ2×ˉ2×ˉ2×ˉ2×ˉ2

{2 + ˉ5 = ˉ3}

5x−4×3x3 = 15xˉ4+3 {am×an = am+n}

b) 42×4−5 f) 3x4×x−6

52÷2−3 = 22−ˉ3 {am÷an = am−n}

Base

Use the Index Laws to simplify each of the following: 22×2−5 = 22+ˉ5 {am×an = am+n}



d3

Index

= 15x−1 c) 24×2−3 g) 3x4×3x−5

= 4x−7 k) 4−2÷42 o) b3÷b−2 Index Law 3

a ÷a = a

(a ) = a

n

m−n

m n

{ˉ4 − 3 = ˉ7} l) 10−5÷10−3 p) 12x5÷4x−3

Index Law 2 m

d) 107×10−5 h) 4x6×2x−2×x−5

8x−4÷2x3 = 4xˉ4ˉ3 {am÷an = am−n}

{2 − ˉ3 = 2+3 = 5} j) 95÷9−4 n) a−5÷a−4

{ˉ4 + 3 = ˉ1}

m×n

Zero Index a0 = 1

3

Use the Index Laws to simplify each of the following: (3−2)4 = 3ˉ2×4 { (am)n = am×n}



= 3−8

(x−2)−3 = xˉ2×ˉ3 { (am)n = am×n}

{ˉ2×4 = ˉ8}

= 26

{ˉ2×ˉ3 = 6}

a) (2−2)3

b) (3−2)−2

c) (x2)4

d) (10−2)−3

e) 44×42

f) x3×x5

g) 106×103

h) 2−5×2

i) 1.33×1.35

j) 105÷102

k) 5x3×2x4

l) (x−4)2×9x3÷2x2

4 Simplify the following expressions: a) 7x − 2x b) 4x + 3x d) 4 + 9x − 3x e) 3d − 4a + 3d g) 2×3x h) 4x×5 − 2 − k) 3y×−2y3 j) 3x × 4x m) 20x÷4 n) −15x÷3 p) 14ab÷4a q) −10x5÷−4x2 5

Expand each of the following:

a) d) g) j)

5(x + 3) 2(4g − 3) − 4(2p − 3) − 3x(4x − 2)

5y − 7y = −2y

ˉ3x×4x = −12x2

c) 2x − 5x f) 4xy2 + 5a5 − 2xy2 + 6a5 i) −5n×2n l) 2x2×5x×2x o) −10b÷−2b r) −12d6e5f ÷ 36d3e3

ˉ3(2y − 5) = −6y + 15

b) 2(h + 3) c) 5(d + 2) e) 8(2x − 3) f) 7(2x − 3) − h) 6(c + 1) i) −x(x + 4) k) −3x(2x − 4y) l) −4x(3y − 2x) 3(x − 2) + 2(x + 5) = 3x − 6 + 2x + 10 = 5x + 4

6

Simplify each of the following by expanding and then collecting like terms:

a) c) e) g) i) k)

3(x + 2) + 5(x + 1) b) 2(y − 3) + 4(y + 2) d) − t(3t + 1) + t(2t − 2) f) (x + 2)(x + 3) h) (x + 4)(x – 1) j) (x + 2)(x – 2) l)

2(x + 3) + 5(x + 1) 4(y − 1) + 3(y + 5) − 5x(3x − 2) + −2(x − 1) (x + 1)2 (x + 1)(x – 1) (2x – 1)(2x – 3) − −

−9x5 − 12x2 = ˉ3x2(3x3 + 4) 7 Factorise each of the following: a) 5x + 10 b) 3x + 6 c) 4x + 16 d) 18a + 3 e) 24r + 6 f) 14y + 7 g) 4x − 12 h) 2y − 4 i) 5x − 15 j) 6p − 10 k) 10w − 8 l) 10ab − 5a m) ˉ5x − 15 n) ˉ2x − 6 o) ˉ2x + 8 q) 14x3 − 21x r) 24x3 − 18x2 p) 10b2 − 12b

Chapter 5 Review 1

67

8 Calculate the area of each of the following composite shapes: a) b) c) 4m

9m

5m

12 m

8m

3.4 m

16 m

15 m

5.8 m

9 Find the surface area of each of the following prisms: a) b) 25 cm

4 cm 9 cm

c)

32 cm

7 cm

20 cm

30 cm

d)

4.7 cm

9 cm 8 cm

5c

m 7c m

7 cm

10 The four sides and the top of the shed are to be given two coats of paint. How much paint is needed for two coats if 1 litre of paint will, on average, cover 15 m2. What is the cost of paint @ $74.00 for 4L? Shed Outside Dimensions Length 12.19 m Width 2.44 m Height 2.59 m 11 Write a rule for the following pattern:

12 Write a rule for the following table: x y

68

1 9

2 4

3 ˉ1

4 ˉ6

5 10 20 ˉ11 ˉ36 ˉ86

13 Write a rule for each of the following graphs: a) b)

y

y

6

7

5

6

4

5

3

4

2

3

1

2 1 -3 -2 -1 -1

1

-3 -2 -1

x

2 3

1

-1

2 3

x

-2 -3

14 Copy and complete each of the following tables and draw a graph of the rule: a) b) x y = 2x − 2

-2

-1

0

1

2

c)

x y = ˉ2x + 4

-2

-1

0

1

2

x y = x2 − 2

-2

-1

0

1

2

d)

x y = x2 − 1

-2

15 Draw a sketch of: a) y = 2x + 3 b) y = 2x − 1 c) y = ˉ2x + 4

-1

0

1

2

A tourist sees a black sheep and says "The sheep are black." The mathematician says "There is at least one sheep, and one side of that sheep is black."

16 Sketch x2 + y2 = 5. x

x +y =5 2

2

ˉ2

ˉ1

0

1

2

1 or ˉ1

2 or ˉ2

5 or ˉ 5

2 or ˉ2

1 or ˉ1

1 -2

3 4

Exercise 5.3 Mental computation 1 Spell Parabola. 2 What is the linear rule for:

x y

0 -5

2 1

3 Roughly sketch the rule: y = 2x + 1 4 What is the formula for the area of a circle? 5 What is the area of a triangle, height = 5 m and base = 8 m? 6 Simplify: 9x − 7x 7 Expand: 3(x − 2) I'm a great believer in luck, and I find 8 Factorise: 6x + 8 the harder I work the more I have of it 9 105×103 Thomas Jefferson. 10 24×15 Chapter 5 Review 1

69

Review 2 Exercise 5.4 1 Write each of the following in index form: 3×3×3×3 = 34

b) 6×6×6×6×6 e) 9.2×9.2×9.2×9.2

= 2−3

a) 24×2−2 e) x7×x−3

= 25

i) 33÷3−6 m) x−4÷x8 Index Law 1 am×an = am+n

3

{2 + ˉ5 = ˉ3}



b) 53×5−7 f) 4x5×x−3

= 15x−1 c) 34×3−2 g) 2x2×4x−5

{2 − ˉ3 = 2+3 = 5} j) 65÷6−4 n) a−3÷a−4

d) 108×10−5 h) 4x3×3x−2×x−5

m−n

{ˉ4 − 3 = ˉ7} l) 10−7÷10−3 p) 9x4÷3x−3

Index Law 3

a ÷a = a n

= 4x−7 k) 2−5÷22 o) b3÷b−2

Index Law 2 m

{ˉ4 + 3 = ˉ1}

8x−4÷2x3 = 4xˉ4ˉ3 {am÷an = am−n}

m×n

(am)n = a

Zero Index a0 = 1

Use the Index Laws to simplify each of the following: (3−2)4 = 3ˉ2×4 { (am)n = am×n}



c) 10×10×10×10×10 f) ˉ4×ˉ4×ˉ4×ˉ4×ˉ4

5x−4×3x3 = 15xˉ4+3 {am×an = am+n}

52÷2−3 = 22−ˉ3 {am÷an = am−n}

Base

Use the Index Laws to simplify each of the following: 22×2−5 = 22+ˉ5 {am×an = am+n}



d3

ˉa×ˉa×ˉa= (ˉa)3

a) 1×1×1 d) x×x×x×x 2

Index

= 3−8

{ˉ2×4 = ˉ8}

(x−2)−3 = xˉ2×ˉ3 { (am)n = am×n}

= 26

{ˉ2×ˉ3 = 6}

a) (2−2)3

b) (3−2)−3

c) (b2)3

d) (10−1)−4

e) 44×43

f) x3×x5

g) 102×103

h) 2−6×21

i) 3.23×3.24

j) 109÷102

k) 5x2×3x2

l) (x−3)2×8x3÷2x2

ˉ3x×4x = −12x2 5y − 7y = −2y 4 Simplify the following expressions: a) 5x−4x b) 8x+3x c) 3x−6x d) 2+8x−2x e) 5b−4b+3b f) 7x2+5y5−2x2+3y5 g) 9×3x h) 3x×7 i) −3b×2b

70

j) −3x2 × −5x m) 12x ÷ 4 p) 14xy ÷ 8x 5

k) 4x × −2x3 n) −12x ÷ 4 q) −6b5 ÷ −4b2

ˉ3(2y − 5) = −6y + 15

Expand each of the following:

a) 4(x + 3) d) 3(2c − 1) g) −2(3x − 4) j) −2x(3x − 1)

l) 5x2 × 3x × 2x o) −12n ÷ −2n r) −36x5y6z ÷ 24x3y3

b) 3(a + 2) c) e) 5(3d − 4) f) − h) 7(x + 4) i) k) −4x(3x − 2y) l)

6(b + 1) 5(3e − 2) − x(4x + 3) − 4x(2y − 4x)

3(x − 2) + 2(x + 5) = 3x − 6 + 2x + 10 = 5x + 4

6

Simplify each of the following by expanding and then collecting like terms:

a) c) e) g) i) k)

2(x + 4) + 5(x + 3) b) 6(x − 2) + 2(x + 3) d) − a(2a + 1) + a(3a − 4) f) (x + 1)(x + 3) h) (x + 4)(x – 1) j) (x + 3)(x – 3) l)

7 Factorise each of the following: a) 5x + 10 b) 3x + 9 d) 18p + 3 e) 12v + 6 g) 4m − 8 h) 2n − 6 j) 6x − 10 k) 12x − 8 m) −5x − 20 n) −2x − 8 2 q) 21x3 − 14x p) 14b − 12b

4(x + 5) + −2(x + 1) − 4(w − 3) + 3(w + 2) − 4b(3b − 2) + −2(b − 1) (x + 1)2 (x + 2)(x – 2) (3x – 1)(2x + 3) −9x5 − 12x2 = ˉ3x2(3x3 + 4)

c) 4x + 8 f) 15w + 5 i) 5o − 20 l) 8xy − 6x o) −2x + 6 r) 24x3 − 12x2

8 Calculate the area of each of the following composite shapes: a) b) c) 6m

9m

6m

13 m

8m

7.4 m

14 m

10 m

6.5 m

9 Find the surface area of each of the following prisms: a) b) 20 cm

3 cm 9 cm

7 cm

28 cm 16 cm

24 cm

Chapter 5 Review 1

71

c)

d)

1.9 cm

8 cm 8 cm

3c

m 4c m

10 Write a rule for the following pattern:

4 cm

11 Write a rule for the following table: x y

1 ˉ7

2 ˉ3

3 1

4 5

5 9

10 29

20 69

12 Write a rule for each of the following graphs: a) b) y 5

1

3

-3 -2 -1 -1

2 1 -3 -2 -1 -1

y

2

4

1

2 3

x

-2

x

2

1

-3 -4

-2

-5

-3

-6

13 Copy and complete each of the following tables and draw a graph of the rule: a) b) x y = 3x − 1

-2

-1

0

1

2

-2

-1

0

1

2

c) x y = x2 + 3

x y = ˉ2x + 3

-2

-1

0

1

2

x y = x2 − 1

-2

-1

0

1

2

d)

14 Draw a sketch of: a) y = 2x + 1 b) y = 2x − 2 c) y = ˉ2x + 1

Before you build a better mousetrap, it helps to know if there are any mice out there - Mortimer B Zuckerman.

15 Sketch x2 + y2 = 9.

72

x

ˉ3

ˉ2

0

2

3

x2 + y2 = 9

0

5 or ˉ 5

3 or ˉ3

5 or ˉ 5

0

Number and Algebra → Real Numbers  Solve problems involving direct proportion.  Explore the relationship between graphs and equations corresponding to simple rate problems.  Understand the difference between direct and inverse proportion, identifying these in real-life contexts and using these relationships to solve problems.

I'm 8.5 heads tall.

A TASK A common problem for artists is drawing in proportion. The basic unit for drawing the human figure is the 'head' - the distance from the top of the head to the chin. Investigate the following human drawing proportions: Average - 7.5 heads tall. Noble - 8 heads tall. Heroic - 8.5 heads tall (bigger chest, longer legs).

A LITTLE BIT OF HISTORY The human figure has been drawn since prehistoric times. The human figure is probably the most difficult subject for an artist. • Ancient Greeks carved human figures in marble. • 1487 - Leonardo da Vinci suggests ideal human proportions with the Vitruvian Man. • 19th century European human figures drawn with clothes. • Drawing the human figure in the 21st century is considered mandatory for an artist.

Leonardo da Vinci used the Golden Proportion in his painting of the Mona Lisa.

73

Warmup 3

A ratio can be written as 3: 10, a fraction , 10 a decimal 0.3, a percentage 30% Exercise 6.1 Write the following comparisons as ratios: 17 people passed the test and 4 failed. a) What is the ratio of pass to fail? b) What is the ratio of fail to pass? c) What is the ratio of pass to the total? d) What is the ratio of fail to the total



The mathematical symbol for ratio is

17 : 4 4 : 17 17 : 21 4 : 21

1

The Maths class has 13 girls and 11 boys. a) What is the ratio of girls to boys? b) What is the ratio of boys to girls? c) What is the ratio of girls to the total number in the class? d) What is the ratio of boys to the total number in the class?

2

Last month there were 24 sunny days and 7 cloudy days. a) What is the ratio of sunny to cloudy days? b) What is the ratio of cloudy to sunny days? c) What is the ratio of sunny to the total number of days in the month? d) What is the ratio of cloudy to the total number of days in the month?

3 Write each of the following ratios as a fraction, a decimal and a percentage:



fraction

decimal

percentage

1:4

1 4

0.25

25%

5:2

5 2

2.5

250%

a) e) i) m)

1 : 5 1 : 10 7 : 10 8 : 10

b) f) j) n)

1 : 2 2 : 5 3 : 10 5 : 5

c) g) k) o)

Make a percentage by multiplying by 100.

1 : 4 2 : 10 6 : 10 9 : 10

d) 3 : 5 h) 4 : 5 l) 5 : 10

4 Write each of the following fractions as a ratio, a decimal and a percentage:

74

1



a) 2



e)

3 4



3

4

1

b) 10

c) 5

d) 4

f) 2 12

g) 3 101

h) 6 53

Warmup Exercise 6.2 Simplify the following ratios: 4.5 : 5.5

15 : 9

15 = 9 3x 5

= 3x 3 5

= 3

=5:3

8b : 4b

4.5 x 10 = 5.5 x 10 45 = 55 5x 9 = 5 x 11

=

9 11

8b

= 4b 8

= 4 =

2 = 2 : 1 1

= 9 : 11

1

2 : 4

2

3 : 6

3

5 : 10

4

3 : 12

5

2 : 8

6

5 : 20

7

2.5 : 1.5

8

2.4 : 1.8

9

5 : 1.5

10 2.8 : 2.1

11 4.0 : 1.2

12 8a : 12a

13 2c : 8c

14

15 1.44y : 7.2y

9x : 15x

Exercise 6.3 Express each of the following as a ratio and simplify 500g to 2 kg = 500 g to 2000 g = 500 : 2000

40 mins to 1 hour 30 mins = 40 mins to 90 mins = 40 : 90

=

=

500 2000 1× 500 = 4 × 500

First make the units the same. Calculators are good at simplifying ratios (Technology 6.1).

40 90 4 ×10 = 9 ×10

=1:4

=4:9

1

200 g to 1 kg

2

600 g to 3 kg

3

500 g to 3 kg

4

2 kg to 100 g

5

1.5 kg to 500 g

6

2.5 kg to 1.5 kg

7

30 mins to 1 hour

8

20 mins to 1 hour

9

50 mins : 2 hours

10 3 hours to 30 mins

11 1.5 hours to 4 hours

12 3 hours to 1 hr 40 mins

13 600 mm to 1.2 m

14 3.5 m to 700 mm

15 500 mm to 2.5 m

16 45 km to 15 km

17 32 km to 48 km

18 4.5 km to 1.5 km

19 50 L to 75 L

20 20 L to 32 L

21 2.4 L to 3.2 L

22 600 mL to 1.2 L

23 2 L to 400 mL

24 1.2 m to 60 cm

25 80 cm to 2.4 m

26 40 cents to $2

27 $3 to 60 cents Chapter 6 Proportion

75

Proportion When two ratios are equal they are said to be in proportion. Exercise 6.4 Which of the following pairs of ratios are in proportion? 4:6 and 10 : 15 4

10

= 15

= 6

2× 2

5× 2

= 2×3 = 5× 3 =2:3 =2:3 The ratios are equal. They are in proportion. 1

4 : 2 and 6 : 3

2

8 : 4 and 10 : 5

3

12 : 6 and 14 : 7

4

6 : 2 and 6 : 3

5

12 : 4 and 9 : 3

6

15 : 5 and 14 : 7

7

5 : 20 and 2 : 8

8

10 : 4 and 15 : 6

9

12 : 8 and 9 : 6

10 2.0 : 0.5 and 8 : 2

11 2.5 : 0.5 and 10 : 2

12 6 : 4 and 1.8 : 1.2

13 1 : 0.5 and 2.1 : 0.7

14 20 mins to 1 hour and 30 mins to 1 hour 30 mins

Travel 120 km in 80 minutes and travel 180 km in 2 hours.

120 km 180 km 120 180

in in

80 mins 120 mins 80

and 120

2:3

and 2:3

A ratio compares quantities of the same kind. Write the quantities with the same units under each other.

The ratios are equal. They are in proportion.

76

15

Travel 200 km in 3 hours and travel 600 km in 9 hours.

16

Travel 90 km in 80 minutes and travel 99 km in 88 minutes.

17

2 tonnes are loaded in 3 hours and 6 tonnes are loaded in 9 hours.

18

Travel 100 km on 8 litres of petrol and travel 200 km on 16 litres of petrol.

19

Travel 80 km on 6 litres of petrol and travel 60 km on 4 litres of petrol.

20

4 kg costs $2 and 6 kg costs $3.

21

4k costs $12 and 3.5 kg costs $10.50.

22

6 parcels of weight 9 kg and 24 parcels of weight 36 kg.

23

$AU200 exchanged for $US204 and $AU500 exchanged for $US510

24

120 cm tall at 12 years and 160 cm tall at 20 years.

Proportion Proportion means that two ratios are equal.

Proportion

Proportion 2:3 = 4:6

a

c

If b = d

2 4 = 3 6

or 2×6 = 3×4

a b

c d

then ad = bc or bc = ad

Cross-multiplication is useful in solving proportion problems.

a b

c d

Exercise 6.5 Assuming proportionality, solve the following problems. If a farmer can grow 132 tonnes of corn If one Australian dollar, $AU1, can be on 30 hectares, how many hectares is exchanged for $US1.03, how many Australian dollars can be exchanged for needed to grow 1500 tonnes of corn? $US236? $AU1 for $US1.03 x for $US236

x×1.03 = 1×236 1× 236 1.03

132 t for 30 ha 1500 t for x ha

{cross multiply}

x×132 = 1500×30 {cross multiply} 1500 × 30 {÷ 132



x =

{÷ by 1.03}



x =



x = $229.13 {calculator}



x = 341 ha {calculator}

by 132}

1

If $AU1 can be exchanged for $US1.06, how many Australian dollars can be exchanged for $US1280?

2

The wheel on an electric motor completes 1800 revolutions in 3 minutes. How many revolutions will it complete in 8 minutes?

3

A car uses 7 litres of petrol to travel 100 km. How much petrol is needed to travel 325 km?

4

A car uses 6.8 litres of petrol to travel 100 km. How far will the car travel on 52 litres of petrol?

5

The design requires a small gear to large gear ratio of 1 : 3. If the large gear has 45 teeth, how many teeth on the small gear?

6

100 g contains 2.3 g of fat. How much fat in 250 g?

7

The concrete requires a mix of 6 bags of sand to one bag of cement. How many bags of cement need to be mixed 35 bags of sand?

8

10 mL of custard powder should be added to 500 mL of milk to make custard. How much custard powder should be added to 2 L of milk?

9

If it takes 4 minutes to run 800 m, how long will it take to run 4 km?

10 If 40 ha of land is valued at $450 000, what will be the valuation of 180 ha? Chapter 6 Proportion

77

Direct Proportion Direct proportion means an increase in one quantity will cause a similar increase in another quantity.

Direct proportion test. Double one quantity and the other quantity will double.

Exercise 6.6 Draw a graph and write a rule for each of the following: y A pizza scrambled egg recipe 6 uses 6 eggs to serve 4 people. 5

0 0

4 6

4

Eggs

People (x) Eggs (y)

0 eggs serves 0 people.

3 2 1

x 0

1

2

3

4

People

5

6

y = mx + c Increase of 6 for 4 steps so m = 6/4 = 1.5 c = 0 {cuts y axis at 0} Eggs = 1.5×people 0 distance, 0 fuel.

78

Distance (x) Fuel (y)

0 0

40 8

1

For each 40 km the rally car uses 8 litres of fuel.

2

For each 60 km the truck uses 12 litres of fuel.

3

15 mg of children's Dosumedrol for every 10 kg.

4

It was suggested that 75 kg of lime per 5 hectares would reduce the soil acidity.

5

The 250 ha paddock yielded 375 tonnes of cotton.

6

The price for cotton was 450 cents for 1 kg.

7

Sound can travel 3.4 km in 10 seconds.

Increase 12 for 60 m = 12/60 c=0 m = 0.2 ∴ Fuel = 0.2×Distance.

Direct Proportion because: 3.4×2 km in 10×2 seconds. 6.8 km in 20 seconds works.

Direct Proportion Exercise 6.7 Solve the following Direct Proportion problems by: a) Finding the rule. b) Using the rule to solve the problem. Mark was paid $30 for picking 12 buckets of tomatoes. How many buckets would Mark need to pick for $20?

When pay = $20 20 = 2.5×buckets 20÷2.5 = buckets 8 = buckets

y

40

Pay ($)

y = mx + c Increase of 30 for 12 steps so m = 30/12 = 2.5 c = 0 {cuts y axis at 0} y = 2.5x {or pay = 2.5×buckets}

30 20 10

x 0

2

4

6

8

10

Buckets

12

{inverse of × is ÷}

1

Jess picked 19 buckets of tomatoes in two hours. How many buckets of tomatoes would Jess be expected in 7 hours?

2

The car can travel 100 km on 6 litres of petrol. How far will the car travel on 25 litres of petrol?

3

Aaron can run 4 km in 20 minutes. How far can Aaron be expected to run in 1.5 hours?

4

Megan can run 3 km in 14 minutes. How long would it take Megan to run 10 km?

5

The infusion pump was set to give 300 mL of medication over 120 minutes. How much medication would be given in 15 minutes?

6

The gravy recipe suggests 2 tablespoons of flour for enough gravy for 8 people. How many tablespoons of flour is needed to serve 50 people?

7

If the electricity tariff is 18 cents per kilowatt-hour, what is the cost for using 65 kilowatt-hours (kWh) in one day?.

8

If a normal household uses 5200 kWh of electricity in four months, how much energy would be used, on average, in 1 week?

30g butter 2 tbsp flour 3 cups chicken stock 1 tbsp sherry

Chapter 6 Proportion

79

Inverse Proportion Inverse proportion means an increase in one quantity will cause a similar decrease in another quantity.

Inverse proportion example. 10 people can build a house in 24 days. 20 people ................................ 12 days. 30 people ................................ 8 days. 40 people ................................ 6 days. People Days 10 24 20 12 10×24 = 20×12

People Days a c b d ac = bd

Exercise 6.8 Assuming inverse proportion, solve the following problems. If 20 people can build a house in 30 It will cost $30 per person if there are days, how long would it take 25 people 45 people on the charter bus. If there to build the same house? are 40 people how much will it cost? 20 people 30 days $30 45 people 25 people x days $x 40 people

20×30 = 25×x 20×30÷25 = x 24 = x

25 people would take 24 days to build the house.

80

30×45 = x×40 30×45÷40 = x 33.75 = x If there are 40 people, it will cost $33.75 per person.

1

If 5 people can build a house in 20 days, how long would it take 20 people to build the same house?

2

If 12 people can build a house in 15 days, how many people will it take to build the same house in 10 days?

3

It will cost $120 per person if there are 15 people on the charter plane. If there are 10 people on the plane what would be the cost per person?

4

It will cost $90 per person if there are 20 people on the charter bus. How many people would be needed to reduce the cost per person to $60?

5

Travelling from A to B will take 30 mins at an average speed of 80 km/h. How long will it take at an average speed of 100 km/h?

6

Travelling from A to B will take 30 mins at an average speed of 80 km/h. What average speed would be needed to reduce the time to 20 mins?

7

If it takes 24 hours for a hose with a water flow of 15 litres per minute to fill a tank, how long will it take to fill the tank if the water flow is increased to 20 litres per minute?

Money and Proportion Proportion

Proportion 2:3 = 4:6 Cross-multiplication is useful in solving proportion problems.

a

2 4 = 3 6

a b

c d

c

If b = d

or 2×6 = 3×4

a b

c d

then ad = bc or bc = ad

Exercise 6.9 Assuming proportionality, solve the following problems. If $AU10 969 can be exchanged for If 2.4 metres of garden fencing costs $68.40, what is the cost of 10.6 metres? MYR35 420 (Malaysian Ringgits), how many Malaysian Ringgits can be 2.4 m costs $68.40 exchanged for $AU45 000? 10.6 m costs x $AU10 969 for MYR35 420 2.4×x = 10.6×68.40 {cross multiply} $AU45 000 for x 10.6 × 68.40 x = {÷ by 2.4} 2.4 10 969×x = 45 000×35 420 x = $302.10 {calculator} 45 000 × 35 420 x = {÷ 10969} 10 969

x = MYR145 310

1

1.6 kilograms of rump steak cost $32.60. a) What is the cost of 2.3 kg? b) How much steak for $15?

2

45 kg of garden mulch costs $68.50. a) What is the cost of 130 kg? b) How much garden mulch for $100?

3

1.55 square metres of cloth cost $8.35. a) What is the cost of 4.8 square metres of cloth? b) How much cloth for $20?

4

If $NZ3610 can be exchanged for JPY238 937 (Japanese Yen). a) How many New Zealand Dollars can be exchanged for JPY1000? b) How many Japanese Yen can be exchanged for $NZ1000?

5

1000 size 23/15 staples cost $12.78. a) How many staples for $100? b) What is the cost of 5000 staples?

6

If $SG967 (Singapore Dollar) can be exchanged for $AU733.22. a) How many Singapore Dollars can be exchanged for $AU1000? b) How many Australian Dollars can be exchanged for $SG1000?

Direct proportion means an increase in one quantity will cause a similar increase in another quantity.

Direct proportion test. Double one quantity and the other quantity will double.

Chapter 6 Proportion

81

Mental Computation The better you become at mental athletics the better you can think.

Exercise 6.10 1 Spell Proportion. 2 Simplify 15 : 10 3 A car uses 7 litres to travel 100 km. How much petrol is needed to travel 300 km? 4 10 people build a house in 50 days How many can build the house in 20 days? 5 What is the linear rule for: 6 7 8 9 10

x y

0 -4

1 -1

2 2

3 5

Roughly sketch the rule: y = 2x + 3 What is the formula for the circumference of a circle? Simplify: 7a − 5a Expand: 3(x + 2) Patient: I swallowed a clock some time ago. Factorise: 5x + 10 Doctor: Why didn't you see me earlier? Patient: I didn't want to alarm you.

Exercise 6.11 1 Spell Inverse. 2 Simplify 8 : 12 3 A car uses 8 litres to travel 100 km. How much petrol is needed to travel 400 km? 4 20 people build a shed in 5 days How long will it take 4 people to build the shed? 5 What is the linear rule for: 6 7 8 9 10

x y

0 6

1 4

Roughly sketch the rule: y = 2x − 1 What is the formula for the area of a circle? Simplify: 4a − 6a Expand: 5(x − 2) Factorise: 6x + 8

Exercise 6.12 1 Spell Cross-multiplication. 2 Simplify 15 : 9 3 A car uses 6 litres to travel 100 km. How much petrol is needed to travel 150 km? 4 20 people build a house in 30 days How many can build the house in 10 days? 5 What is the linear rule for: 6 7 8 9 10

82

x y

0 -2

1 2

2 2

3 0

Inverse Land? What is 7 times 6? What is 6 times 7?

2 6

Roughly sketch the rule: y = ˉ2x + 4 What is the formula for the circumference of a circle? Simplify: 8a − 3a Expand: 6(x + 3) Factorise: 6x + 10

3 10

42 24

Competition Questions

Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 6.13 1

A pedestrian is walking at a speed of 6 km/h. How far does the pedestrian walk in 10 minutes?

Hint: Make the units the same.

2

A person is running at a speed of 12 km/h. How far does the person run in 20 minutes?

3

A car is travelling at a speed of 80 km/h. How far does the car travel in 10 seconds?

6 km in 60 mins x km in 10 mins 6×10 = x×60

4

Mowing is quoted at $93.50/ha. What would be the cost of mowing 1 km2?

5

Harvesting standing coarse grain is quoted at $186.50/ha. What would be the cost of harvesting 450 acres (1 acre = 0.405 hectares)?

6

At idle speed, the crankshaft of an engine typically rotates at 800 revolutions per minute? How many degrees does the crankshaft turn in 1 second?

7

A fan spins at 1200 revs per minute. In one second, how many degrees does the fan turn?

8

Karen can pick 12 buckets of tomatoes in 1 hour. Megan can pick 10 buckets in 40 minutes and Eun-Young can pick 8 buckets in 30 minutes. How long would it take the three people to pick 100 buckets of tomatoes?

9

Matthew can pick 14 buckets of tomatoes in 1 hour. Adam can pick 10 buckets in 50 minutes and Aaron can pick 9 buckets in 30 minutes. How long would it take the three people to pick 100 buckets of tomatoes?

10

Pipe A can fill a tank in 120 minutes. Pipe B can fill the tank in 180 minutes. Pipe C can fill the tank in 90 minutes. If the three pipes are used together, how long will it take to fill the tank?



6 ×10 =x 60

Walk 1 km in 10 mins

800×360° in 60 secs x° in 1 sec 800×360×1 = x×60

800 × 360 60

=x

Rotates 4800° in 1 second.

Karen: 12 buckets in 1 h. Megan: 15 buckets in 1 h. Eun-Y: 16 buckets in 1 h. Together: 43 buckets in 1 h. 100 buckets in x.

Actuaries analyse all kinds of data. For example, actuaries may design life insurance policies and premiums based on calculations involving economic trends, unemployment, illness, accident, and death probabilities. An actuary is a kind of super mathematician and is paid accordingly. • Relevant school subjects are English, Mathematics, Science. • Courses normally involve a University degree and post graduate study. Chapter 6 Proportion

83

Investigations

C

Investigation 6.1 Direct Proportion 1 Draw a circle. 2 Mark two points, A and B, on the circumference. A 3 Join A and B to the centre O. 4 Mark C on the circumference, and join A and B to C. 5 Measure angles AOB and ACB. 6 What do you notice? Does the size of the circle make a difference? Investigate why?

O

B C O

B

A

Circle ∠AOB ∠ACB ratio

1

Investigation 6.2

2

3

4

6

Inverse Proportion

The graph of Direct Proportion is a straight line. 0 0

y

6

6 4

5 4

Eggs

Eggs (x) People (y)

5

0 eggs serves 0 people.

3 2 1

x 0

Investigate

1

2

3

4

People

5

6

The graph of Inverse Proportion?

Inverse proportion example. 10 people can build a house in 24 days. 20 people ................................ 12 days. 30 people ................................ 8 days. 40 people ................................ 6 days.

84

A Couple of Puzzles Exercise 6.14 1 Desley has been complaining that she paid $18 500 in tax last financial year. If her tax rate was 31.5%, what was Desley's nett income last financial year? 2

What is the secret number? • It is a fraction. • The decimal equivalent has one decimal place. • The numerator is a square number. • The denominator is between 10 and 20. • The fraction is between 0.4 and 0.8

A Game Estimate. Players, or teams, take turns in estimating the average of a set of numbers. Sets of numbers are written on cards. 1 A card is selected at random. 28 18 The player, or team, estimates the 20 47 13 49 38 8 average of a set of numbers 33 37 48 written on the card. 9 16

2

The estimate is scored:

3

Highest score, after five or ten estimates, wins.

Within 1 of the correct answer: Within 2 of the correct answer: Within 3 of the correct answer: Within 4 of the correct answer: Within 5 of the correct answer:

48 7

6 35 34

45 41 46 18 12 24

5 points 4 points 3 points 2 points 1 point

A Sweet Trick 1 2 3 4

Show your audience your 9 times hand calculator Ask your audience to choose a multiplicant from 1 to 9. Bend over the fourth finger from the left The fingers left give the answer.



3 fingers one side, 6 fingers the other = 36 23

1

1

4 9×4 = 36

45 23

6

Try: 3×9, 7×9, 5×9 etc.

Chapter 6 Proportion

85

Technology Technology 6.1

Simplifying Ratios

b Calculators, with a c , are very good at simplifying ratios:

1

6 9



6

a bc

9

=

2r3

meaning 3

2

36 60

36

a bc

60

=

3r5

meaning 5

Technology 6.2

2

3

Direct Proportion

Setup a spreadsheet to automatically solve direct proportion problems.

1 2 3 4 5

A $AU 1 x

B for for

C $US 1.03 236

x

=

$229.13

If one Australian dollar, $AU1, can be exchanged for $US1.03, how many Australian dollars can be exchanged for $US236? $AU1 for $US1.03 x for $US236

Enter formula: =a2*c3/c2

Technology 6.3

x×1.03 = 1×236



x



x

Inverse Proportion

Use a spreadsheet to graph Inverse Proportion. 1 Enter points (product = 240). 2 Graph points.

1 2 3 4 5

A B People Days 5 48.0 10 24.0 15 16.0 20 12.0

1× 236 = 1.03

{cross multiply} {÷ by 1.03}

= $229.13 {calculator}

Inverse proportion example. 10 people can build a house in 24 days. 20 people ................................ 12 days. 30 people ................................ 8 days. 40 people ................................ 6 days.

60.0

Enter formula: =240/a2

50.0 40.0 30.0 20.0 10.0 0.0 0

20

40

60

80

Technology 6.4 Proportion Games Search the Internet for some of the many Proportion games and applets. Use search phrases such as: 'proportion game applet', 'inverse proportion applet'.

86

100

Chapter Review 1 Exercise 6.15 1 Simplify the following ratios: a) 3 : 15 b) 6 : 8

c) 15 : 20

d) 12 : 18

f) 5 : 1.5

2

e) 2.4 : 1.8

Which of the following pairs of ratios are in proportion?

a) 4 : 2 and 10 : 5

b) 12 : 4 and 9 : 3

c) 12 : 6 and 14 : 7

d) Travel 200 km in 3 hours and travel 600 km in 9 hours. e) 2 tonnes are loaded in 5 hours and 6 tonnes are loaded in 15 hours. 3

Draw a graph and write a rule for each of the following:

a) For each 40 km the rally car uses 10 litres of fuel. b) Sound can travel 3.4 km in 10 seconds. 4

Assuming proportionality, solve the following problems.

a) If $AU1 can be exchanged for $US1.08, how many Australian dollars can be exchanged for $US1280? b) Aaron can run 4 km in 20 minutes. How far can Aaron be expected to run in 50 minutes? c) A car uses 6.8 litres of petrol to travel 100 km. How far will the car travel on 52 litres of petrol? 5

Assuming inverse proportion, solve the following problems.

a) If 5 people can build a house in 20 days, how long would it take 20 people to build the same house? b) It will cost $50 per person if there are 20 people on the charter bus. How many people would be needed to reduce the cost per person to $40? c) Travelling from A to B will take 30 mins at an average speed of 80 km/h. What average speed would be needed to reduce the time to 20 mins? From the graph, find the litres per 100 km fuel consumption for each vehicle. y

a)

25

Litres (L)

6

b) c)

20 15 10 5

x 0

50

100 150 200 250 300

Distance (km)

Chapter 6 Proportion

87

Chapter Review 2 Exercise 6.16 1 Simplify the following ratios: a) 10 : 5 b) 6 : 4

c) 12 : 8

d) 20 : 15

f) 3 : 1.5

2

e) 1.2 : 1.5

Which of the following pairs of ratios are in proportion?

a) 6 : 2 and 9 : 3

b) 5 : 4 and 10 : 2

c) 15 : 5 and 12 : 4

d) Travel 300 km in 4 hours and travel 450 km in 6 hours. e) 6 tonnes are loaded in 4 hours and 10 tonnes are loaded in 6 hours. 3

Draw a graph and write a rule for each of the following:

a) For each 50 km the rally car uses 10 litres of fuel. b) The 250 ha paddock yielded 325 tonnes of cotton. 4

Assuming proportionality, solve the following problems.

a) If $AU1 can be exchanged for $US0.89, how many Australian dollars can be exchanged for $US750? b) Aaron can run 4 km in 20 minutes. How far can Aaron be expected to run in 2 hours? c) A car uses 7.3 litres of petrol to travel 100 km. How far will the car travel on 64 litres of petrol? 5

Assuming inverse proportion, solve the following problems.

a) If 10 people can build a house in 30 days, how long would it take 25 people to build the same house? b) It will cost $65 per person if there are 30 people on the charter bus. How many people would be needed to reduce the cost per person to $57? c) Travelling from A to B will take 40 mins at an average speed of 90 km/h. What average speed would be needed to reduce the time to 30 mins? 6

From the graph, find the litres per 100 km fuel consumption for each vehicle. y

a)

Litres (L)

25

15

c)

10 5

x 0

88

b)

20

50

100 150 200 250 300

Distance (km)

Measurement and Geometry  Pythagoras and Trigonometry  Investigate Pythagoras’ Theorem and its application to solving simple problems involving right-angled triangles.  Understand that Pythagoras’ Theorem is a useful tool in determining unknown lengths in right-angled triangles and has widespread applications.  Recognise that right-angled triangle calculations may generate results that can be integral, fractional or irrational numbers known as surds.

If my squares were made of gold. Would you choose the large square or the two smaller squares?

A TASK A popular puzzle is the Pythagorean Puzzle. • Research the puzzle. • Make your own pieces. • Demonstrate that you can make and solve your own puzzles.

A LITTLE BIT OF HISTORY 1900-1600 BC A Babylonian Tablet contains the Theorem. 560-480 BC Pythagoras' Theorem: In a right-angle triangle c2=a2+b2. 300 BC Euclid supplies two different proofs and states the converse that if a2+b2=c2 then a right-angled triangle. Present day Hundreds of proofs exist. The Theorem has thousands of applications.

89

Pythagorean Triads Early civilisations knew that the following ratios produced right-angled triangles:

3:4:5 5:12:13

And that is how buildings were built square.

Exercise 7.1 1 Use the 3:4:5 ratio to make a right-angled triangle. a) Make a loop of string with knots at 3, 4, 5 intervals. b) Hold tight at the knots to make a triangle with sides 3, 4, 5. c) Use a protractor to check the right-angle. 3 4 5

4

5 3

2 Use the 6:8:10 ratio (double 3:4:5) to make a right-angled triangle. a) Make a loop of string with knots at 6, 8, 10 intervals. b) Hold tight at the knots to make a triangle with sides 6, 8, 10. c) Use a protractor to check the right-angle. 6 10

8

10

8

6 3 Use the 5:12:13 ratio to make a right-angled triangle. a) Make a loop of string with knots at 5, 12, 13 intervals. b) Hold tight at the knots to make a triangle with sides 5, 12, 13. c) Use a protractor to check the right-angle. A few Pythagorean Triads 3:4:5 5 6:8:10 9:12:15 12 13 5:12:13 13 12 10:24:26 15:36:39 8:15:17 16:30:34 5

90

Pythagoras' Theorem In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

c2 = a2 + b2

c

a b

5 = 25 2

42 = 16

c2 = 52 = 25 2 2 a + b = 32 + 42 = 9 + 16 = 25 2 Thus c = a2 + b2 The triangle is right-angled.

32 = 9 Exercise 7.2 1 Copy and complete the following table: a=3 a=4 a=6 a=5 a=7

b=4 b=5 b=8 b=12 b=15

a +b =25 a2+b2=41 a2+b2= a2+b2= a2+b2= 2

2

c=5 c=6 c=10 c=13 c=16

c =25 c2=36 c2= c2= c2= 2

Right-angled? yes. c2 = a2 + b2 no. c2 ≠ a2 + b2

2 Which of the following triangles are right-angled triangles? a) b) c) 24

25 7

15

10

26

24

10

10

a2 + b2 = c2 al c2 a2 + bg2on= a a2 d+i b2 = c2 a2 + b2 = c2

3

A rectangular picture frame measures 58 cm by 67 cm with a diagonal of 88.6 cm. Is the picture frame square?

4

A carpenter needs to know whether a door frame is square. If the door frame is square then a door can be fitted. The door frame measures 810 mm by 2000 mm and the diagonal is 2140 mm.

5

A 2.3 m ladder is leaning against a wall. The bottom of the ladder is 0.8 m from the wall and the top of the ladder is 2.16 m up the wall. Is the wall vertical?

6

A rectangular gate measures 1.2 m by 2.3 m with a 2.4 m diagonal. Is the gate square? If not, should the diagonal be longer or shorter? Chapter 7 Pythagoras' Theorem

91

Hypotenuse c2 = a2 + b2 In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

c

a b

The hypotenuse is the longest side. It is opposite the right-angle (90°). Exercise 7.3 1 Find the length of the hypotenuse in each of the following: First add a, b, c c2 = a2 + b2 2 c=? c = 532 + 472 ? 53 a=53 2 c = 5018 c = 5018 47 b=47 c = 70.84 a) 63

? 34

92

b)

c) ?

26

?

3.7

10

3.5

2

A farmer wishes to check that the gate is square. What should be the length of the diagonal if the gate is 3.4 m by 1.2 m?

3

The window frame needs to be square to accept a 844 mm by 1173 mm window. What should the length of the diagonal of the window frame?

4

The diagonal of a TV screen describes the size of the TV. For example, a 68 cm TV has a screen with a 68 cm diagonal. Calculate the size of each of the following TV screens: a) 54 cm by 41 cm b) 39 cm by 31 cm c) 13.7 inches by 10 inches (Computer monitor)

5

A ladder lies against a vertical wall. The bottom of the ladder is 2.1 m away from the wall, and the top of the ladder is 5.6 m up the wall. 0° What is the length of the ladder?

6

A plane travels from A to B on a bearing of 90° at a speed of 150 km/h for 30 mins. The plane then travels from B to C on a bearing of 180° at a speed of 200 km/h for 90 mins. What is the distance between A and C?

270°

90° 180°

The Shorter Sides In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

a2 + b2 = c2

c

a b

Exercise 7.4 1 Find the length of the unknown in each of the following: First add a, b, c a2 + b2 = c2 2 6.4 c=6.4 a + 5.12 = 6.42 a2 = 6.42 − 5.12 2 b=5.1 a = 14.95 5.1 ? a=? a = 14.95 a = 3.87 a)

?

b)

9 11

c)

6.2

?

7.8

743

1509

? 2

A 2.8 m ladder is to be laid against a wall so that the top of the ladder is 2 m up the wall. How far out from the base of the wall should the ladder be placed? ?

Rounding? See Technology 7.1

3

Ridge Board

Co

r

Calculate the length of the ceiling joist.

on

te af

m

R

m

om

C

1.4 m

m

on

Ra

2.3 m

fte

r

Ceiling Joist Wall Stud

Wall Stud

4

How far up a building will a 25 m ladder reach if the base of the ladder must be 4 m out from the base of the wall?

5

A kite on a 75 m length of string is vertically above a point 22 m from the person flying the kite. How high is the kite? Chapter 7 Pythagoras' Theorem

93

Length of a line Exercise 7.5 Find the length of AB A (-3, 3) and B(4, -2)

y 5

AC = 5 and CB = 7

4 3

A

AB2 = AC2 + CB2 = 52 + 72 = 74 AB = 74 or 8.60

2 1

-5 -4 -3 -2 -1 0 -1

C

-4

y 5

4

4

3

3

2

C

1 1 2 3

4 5

6

x

A

-3

B

-4

4

4 3 2 4 5

-2 -4

x

-3

2

-3

6

-2

3

1 2 3

B

4 5

5

A

-5 -4 -3 -2 -1 0 -1

1 2 3

y

1

C

6

x

A

1

-5 -4 -3 -2 -1 0 -1

1 2 3

4 5

6

x

-2

B C

Plot the points first.

94

1

-4

3 4 y 5

2

-5 -4 -3 -2 -1 0 -1

-2

C

x

6

B

-3

5

-5 -4 -3 -2 -1 0 -1

4 5

-2

1 2 y A

1 2 3

-3 -4

B

Rounding? See Technology 7.1

5

A(3,2), B(6,5)

6

A(1,1), B(4,5)

7 A(-4,3), B(4,2)

8

A(5,2), B(-5,-3)

9

A(-3,-4), B(5,4)

10 A(-1,5), B(2,-3)

11 A(1,1), B(5,5)

12 A(1,2), B(4,6)

13 A(-4,2), B(4,2)

14 A(5,0), B(0,5)

15 A(-3,3), B(2,3)

16 A(2,0), B(2,-3)

Our Number System Real Numbers

Complex Numbers

Rational Numbers

Irrational Numbers

Can be expressed as a ratio a/b where a and b are integers.

Cannot be expressed as a ratio a/b (a and b are integers).

Integers. ˉ3, 5, ˉ18, 0, 3, 9

Surds. √2 = 1.4142135... √3 = 1.7320508...

Fractions. 1/2, 3/4, 5/3, 6/7

Special numbers. π = 3.1415926... e = 2.7182818...

a + bi a and b are real numbers. i is the imaginary part.

Decimals. (Either terminate or recurr.)

Exercise 7.6 Calculate the unknown and describe the result as rational or irrational (If rational describe the result as integral, fractional, or decimal): First add a, b, c c2 = a2 + b2 2 c=? c = 532 + 472 ? 53 a=53 2 c = 5018 c = 5018 47 b=47 The result is irrational. 1 2 3 ?

8

?

43

2.4

28

6

3

?

4 5 6 ?

5.2 3.9

10.5

8.4 ?

5.1

4.5 ?

Chapter 7 Pythagoras' Theorem

95

Mental Computation Exercise 7.7 1 Spell Pyhtagoras. 2 What is Pythagoras' Theorem? 3 Is {6,8,10} a Pythagorean triad? x 4 Find x 8 5 Simplify 5 : 10 6 6 Karen can walk 2 km in 20 mins. How long will it take her to walk 5 km? 7 The trip to work takes 30 mins at 80 km/h. How long will it take at 60 km/h? 8 Simplify: 7a − 5a 9 Expand: 3(x + 2) 46×5 = 46×10÷2 10 46×5 = 460÷2

You need to be a good mental athlete because many everyday problems are solved mentally.

= 230

Exercise 7.8 1 Spell Hypotenuse. 2 What is Pythagoras' Theorem? 3 Is {9,12,15} a Pythagorean triad? 15 9 4 Find x 5 Simplify 8 : 6 x 6 Seb can run 3 km in 15 mins. How long will it take him to run 5 km? 7 The trip to work takes 40 mins at 80 km/h. How long will it take at 100 km/h? 8 Simplify: 9x − 7x 9 Expand: 5(x − 2) Why are mathematicians afraid of 10 64×5

driving a car? The width of the road is negligible compared to its length.

Exercise 7.9 1 Spell Theorem. 2 What is Pythagoras' Theorem? 3 Is {12,16,20} a Pythagorean triad? 20 x 4 Find x 5 Simplify 12 : 8 12 6 Jess can walk 5 km in 60 mins. How long will it take her to walk 3 km? 7 The trip to work takes 30 mins at 60 km/h. How long will it take at 90 km/h? 8 Simplify: 6x − 5x + 3x You may be disappointed if you fail, but you 9 Expand: 4(x + 5) are doomed if you don't try - Beverly Sills. 10 36×5

96

Competition Questions Exercise 7.10 1 Find x. a)

Build maths muscle and prepare for mathematics competitions at the same time.

b) x

3

25

15 x

4

Show that 3a, 4a, 5a is a Pythagorean triad. c2 = (5a)2 a2 + b2 = (3a)2 + (4a)2 c2 = 25a2 = 9a2 + 16a2 = 25a2 2 2 2 ∴c = a + b Thus 3a,4a,5a is a Pythagorean triad. 2

Show that 3m, 4m, 5m is a Pythagorean triad.

3

Show that 5a, 12a, 13a is a Pythagorean triad.

4

If p, q, r is a Pythagorean triad, show that ap, aq, ar is also a pythagorean triad.

Find the value of x. In triangle A In triangle B 2 2 2 y + 7 = 8 x2 = y2 + 32 y2 = 15 x2 = 15 + 9 x2 = 24 x = √24 or 2√6 5 Find the value of x. a)

c)

A

B 3

7

10

20

12 x

9

3

8

y

b) 11

x

x

d)

4

11

x

4 7

9

6

x Chapter 7 Pythagoras' Theorem

97

Investigations Investigation 7.1

Proof of Pythagoras' Theorem

In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

c

a

c2 = a2 + b2

b

There are many proofs of Pythagoras' Theorem. Use the Internet to select one of the proofs and demonstrate it to others. Investigation 7.2

Pythagorean Triples

When all three sides of a right-angled triangle are integers, their lengths form a Pythagorean triple. Integers are whole numbers within the set: 3, 4, 5 is a Pythagorean triple. Z = {..... ˉ4, ˉ3, ˉ2, ˉ1, 0, 1, 2, 3, 4, .....} c2 = a2 + b2

c

a b

Pythagorean triple formula Select two integers n and m with n > m. a = n2 − m2 b = 2nm c = n2 + m2

Use the Pythagorean triple formula to generate Pythagorean triples. Investigation 7.3

Shortest Surface Path

Investigate The shortest surface path on a cube

98

Technology Technology 7.1

Square Root

There are at least two ways of finding the square root with a calculator. 1

9.2

2

9.20.5



9.2

9.2

=

3.033150

yx

=

3.033150 which is 3.03 to 2 decimal places.

which is 3.03 to 2 decimal places.

1

x = x 2 = x 0.5

Rounding to two decimal places, first look at the third decimal place: 56.231694 27.01769 1.07276 4.79634216 less than 5 thus 56.23 5 or more thus 27.02

Technology 7.2

less than 5 thus 1.07

If a= 4, b=3

Triangle Solvers

c=5 A = 53.13° B = 36.87° C = 90.00° A Area = 6 Perimeter = 12

The Internet has a considerable number of 'triangle solvers'. Use one of them to solve previous problems.

Technology 7.3

5 or more thus 4.80

B c

a b

C

Pythagoras Spreadsheet

Setup a spreadsheet to solve right-angled triangles. Enter the formula:

1 Technology 7.3

a 3

b 4

c 5

=sqrt(a1^2+b1^2)

Pythagoras' Theorem

Search the Internet for some of the many Pythagoras' Theorem interactive games and applets. These are very useful in understanding Pythagoras' Theorem.

Petroleum Geologists explore the earth's surface to predict possible locations of oil and natural gas. • Relevant school subjects are Mathematics, Chemistry, Physics, English. • Courses usually involve a University Bachelor degree. Chapter 7 Pythagoras' Theorem

99

A Couple of Puzzles Exercise 7.11 1 What is my age? 2

My age is twice the number of faces on a cube plus three times the number of edges on a cube.

The pentagon can be made with this isosceles triangle. What is the size of x?

A Game Box is played on a grid of dots. 1 Players take turns to draw a line to join two adjacent dots. 2 If a turn completes a box then the player is allowed to draw another line to join another two adjacent dots. Mark on your completed box 3 When all dots have been joined, total the number of boxes.

4x 3x

3x

A Sweet Trick Number off your audience from 1 to the number in your audience. While you are not looking or out of the room, have your audience put a rubber band on someone's finger. Ask your audience to: 1 Multiply the number of the person with the string by 2. 2 Add 3. 3 Multiply the result by 5. 4 If the string is on the right hand add 8. If the string is on the left hand add 9. 5 Multiply by 10. 6 Add the number of the finger (The thumb = 1). 7 Add 2. 8 Ask them to tell you the answer. Mentally subtract 222. The remainder gives the answer.

100

Person 5 5×2=10 10+3=13 13×5=65 65+8=73 73×10=730 730+3=733 731+2=735 735 735 − 222 = 513. 5 person number 5 1 right hand (2 left hand) 3 on third finger

Chapter Review 1 Exercise 7.12 1 Which of the following triangles are right-angled triangles? a) b) c) 51

6.3

15

24 2

A rectangular picture frame measures 28 cm by 23 cm with a diagonal of 36 cm. Is the picture frame square?

3

A carpenter needs to know whether a door frame is square. If the door frame is square then a door can be fitted. The door frame measures 810 mm by 2000 mm and the diagonal is 2158 mm.

4 Find the length of the unknown in each of the following: a) b) c) 8 7.6

17

?

7.2

3.4

39

36

45

9.5

?

2143

4600

? 5

A 3.2 m ladder is to be laid against a wall so that the top of the ladder is 2.5m up the wall. How far out from the base of the wall should the ladder be placed? ?

6 Find the length of AB. a)

b)

y

y

5

C

4

A

3 2

2

1

1 1 2 3

4 5

6

x

-5 -4 -3 -2 -1 0 -1

-4

c) A(5,2), B(6,5)

1 2 3

4 5

6

x

-2

-2 -3

B

4 3

-5 -4 -3 -2 -1 0 -1

C

5

-3

A -4

B

d) A(1,-1), B(4,4)

e) A(-4,-3), B(4,3)

Chapter 7 Pythagoras' Theorem

101

Chapter Review 2 Exercise 7.13 1 Which of the following triangles are right-angled triangles? a) b) c) 26

4.5

3.6

15

12

24

2.7

10

12 2

A 2.2 m ladder is leaning against a wall. The bottom of the ladder is 0.7 m from the wall and the top of the ladder is 2.06 m up the wall. Is the wall vertical?

3

A rectangular gate measures 1.4 m by 2.5 m with a 2.79 m diagonal. Is the gate square? If not, should the diagonal be longer or shorter?

4 Find the length of the unknown in each of the following: a) b) c) 6 ?

4.1

8

?

5.3

3600

4200

? Ridge Board

5

Calculate the length of the ceiling joist.

Co

ter

on

m

m

Co

m

f Ra

1.8 m

1.3 m

m

on

Ra

fte

r

Ceiling Joist

A

b)

y

5

5

4

4

3

3

2

C

Wall Stud

Wall Stud

6 Find the length of AB. a) y

-5 -4 -3 -2 -1 0 -1

2

B

1 1 2 3

-2 -3

4 5

6

x

C

A

-4

c) A(3,2), B(6,4)

102

1

-5 -4 -3 -2 -1 0 -1

B 1 2 3

4 5

6

x

-2 -3 -4

d) A(-1,1), B(-4,4)

e) A(-4,3), B(3,0)

Measurement & Geometry - Geometric reasoning  Use the enlargement transformation to explain similarity and develop the conditions for triangles to be similar.  Solve problems using ratio and scale factors in similar figures.

Same shape but different size. Sound similar?

A TASK A fractal is a geometric shape that can be repeatedly split into parts. Each part being similar to the original shape. • • • •

View some online fractal videos. Take part in some online fractal activities. Try some of the fractal generators. Produce your own fractal for the classroom wall.

A LITTLE BIT OF HISTORY 1670 1904 1915 1975

Liebnitz considered repetition of similar shapes. Koch mathematically defined the Koch Curve. Sierpinski constructed the Sierpinski triangle. Mandelbrot used the term fractal to describe shapes consisting of similar shapes.

103

Congruent Triangles Congruent triangles have exactly the same shape and size. They fit exactly on top of each other. The symbol for congruence is

≡

or

Exercise 8.1 Given that the following pairs of triangles are congruent, Correctly name them (angles and sides must match): D



A

A= F B= D C= E

E F

B

If the angles are in matching order then the sides will be in matching order. And vice versa.

∴ABC ≡ FDE

C

1 2 C F

A

B

D

E

F

B A

D

F E

Two triangles are congruent if: ► three sides on one triangle are congruent with matching sides on the other.

SSS

► two sides and the included angle on one triangle are congruent with the two sides and included angle on the other.

SAS

► two angles and a side on one triangle are congruent with the two angles and matching side on the other.

AAS

Two right-angled triangles are congruent if: ► hypotenuse and a side on one triangle are congruent with hypotenuse and matching side on the other.

104

D C

D A

E

C

B

E B 3 4

C

F

A

RHS

Tests for Congruent Triangles Exercise 8.2

Use the tests for congruence to test whether the following pairs of triangles are congruent: A



W

16 cm

16 cm

side AB = side WP B = P {angle inbetween} side BC = side PT

P 95˚

B 95˚

28 cm

28 cm

∴ABC ≡ WPT {SAS}

C

T

B

1 2 26 cm

C

A

26 cm 30˚ 19 cm

F

M

30˚ 19 cm

O

C

10 m

N

D

A

K

13 m

80˚

13 m

A

C

G

5m A

I

60˚

60˚

60˚

60˚

R

7 mm

A

B

100˚

19 m R

C

100˚

18 m

Z

F

A

21 m

13 m

A

30˚

B

13 m

21 m

D

18 m

B

O

P 7 8 19 m

T 5m

3m

5 6 P B

C

3m

Y

13 m

B

7 mm

M

14 m

B

12 m

3 4 C 13 m 80˚

10 m

12 m

14 m

30˚

L

A

4m Q

G

C

B

5m

4m C

5m

K

Chapter 8 Geometry

H

105

Similarity Transformation A similarity transformation may change both position and size but keeps the same shape.

Measure to check that the original image has been doubled in size.

Exercise 8.3 1 Enlarge the original image by a scale factor of two. A'

a) Choose an external point O. b) Draw lines to image and extend same distance





Scale =

A

OA ' OB ' OC ' = = OA OB OC

2 Scale = 1

O

b)

c)

d)

e)

f)

Reduce the above original images using a scale factor of ½.

a) Choose an external point O. b) Draw lines to image and find the midpoints.





C'

C

a)

2

106

B'

B

Scale =

A

A'

OA ' OB ' OC ' = = OA OB OC

1 Scale = 2

C B

B'

C'

O

Similarity Transformation A similarity transformation may change both position and size but keeps the same shape.

Measure to check that the original image has been doubled in size.

Exercise 8.4 1 Enlarge the original triangle by a scale factor of two. 1 2 3

A (-2,2) double to A' (-4,4) B (-1,-2) double to B' (-2,-4) C (3,0) double to C' (6,0)



OA ' OB ' OC ' Scale = = = OA OB OC



Scale =

a)

y 4 3

A

2 1

-5 -4 -3 -2 -1 0 -1

1 2 3

C

C' 4 5

x

6

-2

2 1

B -3 -4

B'

b)

y

y

5

5

4

4

3

A

5

A'

2

C

1

-5 -4 -3 -2 -1 0 -1

A

1 2 3

3 2

C

1 4 5

6

x

-2

B -3 -4

-5 -4 -3 -2 -1 0 -1

1 2 3

-2

4 5

6

x

B

-3 -4

The enlargement should have a similar shape.

c) A(-3,2), B(6,5), C(2,-2)

d) A(1,1), B(4,5), C(-4,2)

2 Enlarge each following triangle by a scale factor of three. a) A(-3,2), B(6,5), C(2,-2)

b) A(-3,-4), B(2,-4), C(2,2)

3 Reduce each following triangle using a scale factor of ½. a) A(-3,2), B(6,5), C(2,-2)

b) A(1,2), B(4,6), C(6,2)

4 Enlarge each following triangle using a scale factor of 1½. a) A(-3,2), B(6,5), C(2,-2)

b) A(-4,-2), B(2,3), C(-3, 3)

Chapter 8 Geometry

107

Similar Triangles Similar triangles have exactly the same shape but not necessarily the same size. The corresponding angles are equal. The corresponding sides have the same scale factor. The symbol for similarity is

~.

Two triangles are similar if: ϕ

ϕ AAA ► The three matching angles are equal. ∞

► The three matching sides are in the same ratio ► Two matching sides are in the same ratio and the included angles are equal

c

a

∞

kc

ka

b

SSS

kb ka

a b

SAS

kb

Two right-angled triangles are similar if: ► The hypotenuse and a matching side are in the same ratio. a

h

kh

ka

RHS

Exercise 8.5 Prove that ABC ~ DEC A D



{common angle} {both 90°} {3rd angle must be equal}

∴ABC ~ DEC {AAA} B

1

C = C B = E A = D

E

C

Prove that ABC ~ DEC D

2

Prove that ABC ~ DEC A

E

B A

B

C D

108

C E

Tests for Similar Triangles Prove that ABC ~ DEC A C

A = D {alternate angle} B = E {alternate angle} BCA = ECD {vertically opposite}

E

∴ABC ~ DEC {AAA}

B D

3

4

Prove that ABC ~ DEC

Prove that DAB ~ DEC D

D

B

C

E

C

A A

E

Prove that ABC ~ EBD

C

BC 18 3 = = BD 6 1

D 6 A

5

AB 12 3 = = EB 4 1

12

8

E

B = B B

4



{common included angle}

∴ABC ~ EBD {SAS} 6

Prove that CAB ~ DEC C

B

Prove that ABE ~ ACD A

10 6

D 6 A

7

5

E

B C

B

3

8

Prove that DEC ~ ABC

E 2

2

D

Prove that ABC ~ DEC A

D 6 C 8 E

6

3

B

4

15 A

D

12

20

E C

16 B

Chapter 8 Geometry

109

Similar Triangles Exercise 8.6 Find the length of the unknown. A



ABC ~ DEC {AAA}

x

AB BC = DE EC x 20 = 6 5 20 x = ×6 5

D 6



B



5

E

C

20



{same scale factor}

{inverse of ÷ is ×}

x = 24

1

2

A

x

A

36

D

D

5

9

B

7

E

B

C

E

28

3

4

A

42

x

x B

B

5

E

C

C

6

D 10

E

D

D 45

x

15

C 11

C x

A

18

60

48

60

35

E

C

A

D

5

8

x

B

9

E

Building Site Managers usually supervise construction sites to ensure the building is proceeding as planned. • Relevant school subjects are Mathematics and English. • Courses usually involve a Certificate or Diploma in building.

110

A

B

Similar Triangles Exercise 8.7 A 1.2 metre stick casts a 0.83 metre shadow at the same time a tree casts a 14.34 m shadow. What is the height of the tree? ABD ~ ECD {AAA} A

AB BD = EC CD



AB 14.34 = 1.2 0.83

E

1

AB =

1.2

B

14.34

C

0.83 D



{same scale factor}

14.34 ×1.2 0.83

{inverse of ÷ is ×}

AB = 20.73 The tree is 21 m high

A 1.5 metre stick casts a 0.62 metre shadow at the same time a tree casts a 5.4 m shadow. What is the height of the tree? A

2

In trying to calculate the width of a wild river, the following diagram was produced. What is the width of the river at AB?

Wild river B 6m

Q2 and Q3 are difficult. Can you solve them?

3

10m

D

C

13.25m

E

The following figures are similar and the ratio of corresponding sides is 3:1. What is the ratio of the perimeter? What is the ratio of the area? Perimeter: 3x

x

x

3x

Perimeter = x+x+x+x Perimeter =3x+3x+3x+3x = 4x = 12x Area = s2 Area = s2 = x2 = (3x)2 = 9x2 a)

ratio =

Area ratio =

12 x = 3 :1 4x

9x 2 x2

= 9 :1

b) 2x

x

x

Chapter 8 Geometry

111

Mental Computation Mental computation can make

Exercise 8.8 problems easier and quicker. 1 Spell Congruent. 2 What does AAA mean? 3 What is the symbol for similar? 4 Scale factor? One side 4, corresponding side 12. 5 Is {3,4,5} a Pythagorean triad? 6 Two sides in a right-angled triangle are 1 and 3. Hypotenuse? 7 Simplify 15 : 12 h2=32+12 8 Tim can run 4 km in 20 mins. 3 h2=9+1 How long will it take him to run 5 km? h = √10 9 Simplify: 9x − 7x + 3x 1 10 I buy a $5.90 magazine with a $10 note, how much change?

A cat has nine tails. Proof: No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails.

Exercise 8.9 1 Spell Similar. 2 What does SSS mean? 3 What is the symbol for similar? 4 Scale factor? One side 3, corresponding side 12. 5 Is {6,8,10} a Pythagorean triad? 6 Two sides in a right-angled triangle are 1 and 2. Hypotenuse? 7 Simplify 20 : 15 8 Jess can run 5 km in 30 mins. How long will it take her to run 3 km? 9 Simplify: 4x − 8x + 2x 10 I buy a $9.10 magazine with a $20 note, how much change?

The greatest mistake you can make in life is to be continually fearing you will make one -Elbert Hubbard.

Exercise 8.10 1 Spell Transformation. 2 What does AAS mean? 3 What is the symbol for similar? 4 Scale factor? One side 8, corresponding side 4. 5 Is {9,12,15} a Pythagorean triad? 6 Two sides in a right-angled triangle are 1 and 1. Hypotenuse? 7 Simplify 12 : 18 8 Megan can run 5 km in 35 mins. How long will it take her to run 4 km? 9 Simplify: 10x − 3x − 8x 10 I buy a $15.30 magazine with a $20 note, how much change?

112

Competition Questions Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 8.11 1 A building casts a 15 m shadow at the same time as Aaron casts a 1.2 m shadow. Aaron is 1.8 m tall. What is the height of the building? 2

If the height of the mother is 145 cm, what is the height of the child?

The dimensions of the rectangular prism is doubled. What has happened to the surface area?

3

2 5

Original Surface Area A = 2(2×3)+2(2×5)+2(3×5) = 62 Doubled Surface Area A = 2(4×6)+2(4×10)+2(6×10) = 248 Ratio of Doubled to Original = 248 : 62 = 4 : 1

3

The dimensions of the rectangular prism is tripled. What has happened to the surface area?

4

The dimensions of the rectangular prism is doubled. What has happened to the volume?

5

The dimensions of the rectangular prism is tripled. What has happened to the volume?

6

The two circles are similar with a scale factor of 3. If the radius of the smaller circle is r cm, what is the area of the larger circle?

7

The two squares are similar with a scale factor of 2. If the perimeter of the smaller square is p, what is the perimeter of the larger square?

8

The two squares are similar with a scale factor of 2 : 3. If the perimeter of the smaller square is p, what is the area of the larger square?

9

The two cubes are similar with a scale factor of 2 : 5. If the side of the smaller square is s, a) what is the volume of the larger cube? b) what is the surface area of the larger cube?

3

2 5

Chapter 8 Geometry

113

Technology Similar figures have exactly the same shape but not necessarily the same size. The corresponding angles are equal. The corresponding sides have the same scale factor.

Technology 8.1 Fractals A fractal is a geometric shape that can be split into parts. Each part being similar to the original shape. a) Draw the first four iterations of the Koch snowflake'

Start with an equilateral triangle Repeat three times: Add triangles a third the size to each side.

Fractals are found in nature. They have applications in soil mechanics, seismology, medicine and artwork.

Each iteration produces smaller similar shapes.

b) Use Internet software to draw iterations of the Koch Snowflake. Use search phrases such as 'Koch Snowflake' with 'applet', 'interactive' etc. c) Use Internet software to produce some of the beautiful Mandelbrot sets and Julia sets. Technology 8.2 Sierpinski Gasket a) Draw the first four iterations of the Sierpinski Gasket.

Start with a triangle Repeat three times: Find the midpoint of each side. Connect them to form a similar triangle. b) Use Internet software to draw iterations of the Sierpinski Gasket. Use search phrases such as 'Sierpinski Gasket' with 'applet', 'interactive' etc.

114

Investigations Investigation 8.1 Enlarge a Cartoon 1 Select a cartoon. 2 Draw grids on the cartoon (6×4?). 3 Draw a larger (6×4?) grid (3 times the size?). 4 Copy each section onto the larger grid.

How much larger is the area?

Investigation 8.2 Use a Pantograph 1 Make or purchase a Pantograph. 2 Use the Pantograph to enlarge a drawing. 3 Are the figures similar? (Check the ratio of corresponding sides.

Construction instructions are on the Internet.

How does it work?

Investigation 8.3

Tessellation

Investigate Shapes that can make larger versions of themselves Equaliteral triangles? Isosceles triangles? Right-angled triangles? Squares? Rectangles? Rhombuses? Kites?

Chapter 8 Geometry

115

A Couple of Puzzles Exercise 8.12 1 How many different numbers can you make using the digits 1, 2, 3? 2

Two friends pressed enough olives to fill an 8 litre barrel. If they only have a 5 litre container and 3 litre container, how can they arrange to have 4 litres each?

5L

1L

A Game Tug of war is a two player game in which each player starts with a total of 50. The winner is the person who moves the marker from the start to a win. 1

Each player secretly writes down a number (Each player then subtracts the number from their total).

2

The player with the highest number moves the marker one place towards their win.

3

The above two steps are repeated.

If your total becomes 0 then you have no moves left.

Start Player A

Player B Win

Win

A Sweet Trick Set up the following trick: 1 Wrap an elastic band around the index finger. 2

Then under and around the middle finger.

3

Loop it onto the index finger again

4

Ask your audience to trap the elastic band by grasping the end of either finger.

You drop the band off the end of the free finger. The band, amazingly, is on the free finger.

116

Chapter Review 1 Exercise 8.13 1 Use the tests for congruence to test 9 cm whether the following pair of triangles are congruent:

B

12 cm

5 4

A

C

-5 -4 -3 -2 -1 0 -1

3

y

4

1

B

V

5 3

9 cm

12 cm

A

2 Enlarge the original triangle by a scale factor of two. a) b) y

2

S

9 cm

C

A

9 cm

T

3

C

2 1

1 2 3

4 5

6

x

-5 -4 -3 -2 -1 0 -1

-2

-2

-3

-3

-4

-4

Prove that ABC ~ ADE

4

1 2 3

4 5

x

6

B

Prove that ABC ~ DEC D

A

B

C C

B

A E

D

E

5 Find the length of the unknown. a) b)

D

A

8 x

E

D

x

12

C

5 B

E

6

C

24

6

A

18

B

A 1.2 metre stick casts a 0.80 metre shadow at the same time a tree casts a 4.5 m shadow. What is the height of the tree?

Chapter 8 Geometry

117

Chapter Review 2 Exercise 8.14 1 Use the tests for congruence to test whether the following pair of triangles are congruent:

C

H 7m

B

50˚

60˚

A

y

y

5

5

4

4 3

3

C

2

1 2 3

4 5

6

1

-5 -4 -3 -2 -1 0 -1

x

C

2

A

1

-5 -4 -3 -2 -1 0 -1

-2

-2

-3

-3

B -4

3

7m

50˚ J

2 Enlarge the original triangle by a scale factor of two. a) b)

A

K

60˚

1 2 3

4 5

x

6

B

-4

Prove that ABE ~ ACD

4

Prove that ABC ~ EBD C

D

10

E

D 5

A

A

C

B

5 Find the length of the unknown. a) b) A

6

3

E

D 45

36

B

E

8

In trying to calculate the width of a wild river, the following diagram was produced. What is the width of the river at AB?

9

x

C

x

E

A

A

Wild river B 8m D

118

B

C 11

D 9

6

B

12m 15.25m

C E

Statistics and Probability  Data representation and interpretation  Identify everyday questions and issues involving at least one numerical and at least one categorical variable, and collect data directly from secondary sources.  Construct back-to-back stem-and-leaf plots and histograms and describe data, using terms including ‘skewed’, ‘symmetric’ and ‘bi modal’.  Compare data displays using mean, median and range to describe and interpret numerical data sets in terms of location (centre) and spread.

Whatdya want? A stem-and-leaf or a histogram?

A TASK Show your class how to use a graphics calculator or Internet applet to help produce a stem-and-leaf plot. • Research graphics calculator and stem-and-leaf plots. • Practice with the problems in Exercise 9.5. • Plan your lesson. • Show your class.

A LITTLE BIT OF HISTORY 1900s Bowley used stem-and-leaf plots for initial data analysis. 1977

Tukey used computer technology of the time to produce stem-and-leaf plots. Stem-and-leaf plots then became popular ways of showing the shape of data and were easy to produce via computer.

Now

Modern computer graphic capability has reduced the use of stem-and-leaf plots.

1 2 3 4 5 6 7

46 0233 1455567 00235677899 2234489 2688 03

3│1 means 31 A stem-and-leaf plot showing symmetrically shaped data.

119

Descriptive Statistics The first step in analysing data is to describe the data using descriptive statistics. The Mean describes the middle of the data.

The mean is also called the average. The mean is heavily affected by extreme scores.

Exercise 9.1 Find the mean of each of the following set of scores: 1, 2, 5, 5, 6 mean =

sum of scores number of scores

mean =

1+ 2 + 5 + 5 + 6 5

1

mean = 3.8

2

3

4

5

6

Balance point is 3.8

1 1, 2, 4, 5, 5, 6 3 1, 1, 2, 2, 2, 3, 4, 4 5 41, 45, 45, 42, 46, 44 7 231, 235, 235, 232, 236, 234 9 4.2, ¯3.6, 1.4, ¯2.8, 3.7,

The Median is the middle of a set of scores.

2 4 6 8 10

1, 2, 5, 5, 6, 130 1, 5, 5, 2, 6, 4 9.1, 9.5, 9.5, 9.2, 9.6, 9.4 ¯1, ¯5, ¯5, ¯2, ¯6, ¯4 ¾, ¼, ½, ¼, ¼, ½

The median ignores extreme high scores and extreme low scores.

Exercise 9.2 Find the median of each of the following set of scores: 2, 7, 5, 1, 3, 5, 7

2, 7, 5, 1, 3, 5, 7, 4

Put the scores in ascending order 1, 2, 3, 5, 5, 7, 7

Put the scores in ascending order 1, 2, 3, 4, 5, 5, 7, 7

Median = 5

Median = 4.5 {Average of 4 & 5}

1 3 5 7

120

{5 is in the middle }

1, 3, 5, 2, 7, 7, 5 51, 53, 55, 52, 57, 57, 55 6.2, 9.8, 3.6, 3.2, 3.1, 3.3 ¯5, 2, 3, ¯2, ¯4, 3

2 1, 3, 5, 2, 7, 7, 500 4 1, 2, 3, 4, 1, 2, 4, 3 6 21, 24, 23, 23, 24, 56 8 1, ¯3, 4, ¯2, ¯1, 2, 1

The Range describes the spread of the data.

The range is the simplest description of the spread of the data.

Exercise 9.3 Find the range of each of the following set of scores: 1, 2, 3, 4, 4 Range = largest − smallest Range = 4 – 1 Range = 3 1 46, 48, 45, 29, 56, 27 3 2.3, 4.7, 3.2, 2.6, 3.5, 2.5 5 632, 635, 636, 632, 631 7 5.2, ¯6.1, 5.8, ¯3.2, ¯4.6

The range is the difference between the smallest and the largest.

2 4 6 8

4, 5, 2, 3, 6, 5, 1 11, 13, 13, 15, 11, 12, 17, 14 ¯2, ¯3, ¯5, ¯4, ¯1, ¯2, ¯6 ¾, ¼, ½, ¼, ¼, ½

Exercise 9.4 Calculate the mean, median, and range for each of the following: If the mean and median differ by a reasonable amount, try to explain the difference. Recent house price sales: $450k, $520k, $480k, $1250k, $530k. mean =

sum of scores number of scores

mean =

450 + 520 + 480 + 1250 + 530 5

mean = $646k

In ascending order: $450k, $480k, $520k, $530k, $1250k Median = $520 Range = 1250 − 450 Range = $800k

The larger price of $1250k has affected the mean. The median is less affected and is probably the better measure of the middle. 1 2

Recent house price sales: $450k, $520k, $480k, $480k, $530k. Recent house price sales: $340k, $420k, $430k, $370k, $420k, $1310k.

The mean is the usual measure of the middle unless there are extreme values.

3 4 5

This week's 400 m times: 92 s, 95 s, 97 s, 93 s, 94 s, 94 s, 88 s Test scores: 7, 8, 8, 7, 6, 9, 9, 10, 7, 8, 10, 10, 6, 8, 8, 9, 7, 9, 9, 8, 10, 6, 8 The ages of the people on the bus: 13, 15, 15, 14, 13, 16, 13, 14, 14, 15, 39 Chapter 9 Statistics

121

Stem-and-Leaf Plots A Stem-and-Leaf Plot is simple, shows the shape of the data and puts the data in order. Exercise 9.5 Use a stem-and-leaf plot to represent the following data. Also find the mean, median, and range. The test scores of 13 students: 45, 37, 46, 48, 39, 42, 45, 42, 23, 37, 45, 46, 41

in order: 23, 37, 37, 39, 41, 42, 42, 45, 45, 45, 46, 46, 48 mean =

sum of scores number of scores

mean =

536 13

Test scores (4|5 is 45)

2 3 3 779 4 122555668

mean = 41.23 median = 42 range = 48 − 23 range = 25

122

1

The test scores of 10 students: 76, 83, 85, 78, 82, 66, 83, 79, 84, 78, 94

2

The Melbourne daily maximum temperature for the last week: 17.0, 18.6, 19.8, 20.2, 18.7, 20.3, 20.6,

3

Number of container ships per month at the Port of Brisbane: 65, 65, 64, 74, 80, 73, 77, 72, 72, 70, 71, 69, 63

4

Number of twenty foot container units per month exported from Flinders Ports in South Australia: 8200, 9500, 11000, 9400, 10000, 9300, 11000, 9600, 7500

5

Ages of Australian Prime Ministers at first taking office: 52, 47, 37, 59, 46, 53, 53, 40, 53, 52, 59, 44, 47, 57, 55, 60, 57, 68, 56, 63, 56, 45, 53, 48, 57, 50, 49

6

The weekly house rents ($): 348, 362, 352, 385, 351, 363, 375, 351, 349, 351, 348, 350, 356, 345, 349 352, 349, 374, 350, 349

Stem-and-Leaf Plots

'Negative skew' The data is or 'Left skew'. 'Symmetrical'. The left tail is longer. mean = median There are few low values.

'Positive skew' or 'right skew'. The right tail is longer. There are few high values

Exercise 9.6 Describe the shape of the following stem-and-leaf plots:

1

Test scores (4|5 is 45)

Exports (9|3 is 93)

7 8 9 10 11 3

2

6 6889 2334 4

Ships (7|2 is 72)

6 34559 7 0122347 8 0

4

7 04567789 022333356677799 038

Test marks (6|6 is 66)

6 7 8 9

There are few low values.

5 2 3456 0 00

Age (5|2 is 52)

3 4 5 6

5

The shape is a negative skew.

2 3 3 779 4 122555668

Rent (35|2 is 352)

34 35 36 37 38 6

5889999 00111226 23 45 5

Temperature (18|6 is 18.6)

17 18 19 20

0 67 8 236

Chapter 9 Statistics

123

Histograms Histograms: Grouped data on the horizontal axis. Frequencies on the vertical axis.

Because Histograms group data, some data value is lost and thus the measures are not as accurate.

Exercise 9.7 Use a stem and leaf plot to represent the following data. Also find the mean, median, and range. Team individual weights: 99, 85, 80, 83, 87, 88, 96, 92, 75, 88, 76, 103, 87, 83, 83, 86, 84, 79, 95.

Weight 70 - 79 80 - 89 90 - 99 100 - 109

Choose groupings so that there are not too many groups, nor too few groups (5 to 12 groups).

No. 3 11 4 1

Each of these groupings are ok. Which do you prefer?

8

Frequency

10

8

Frequency

10 6 4 2 74.5

84.5

Weight 75 - 79 80 - 84 85 - 89 90 - 94 95 - 99 100 - 104

94.5

Weight (kg)

104.5

6 4 2 77

82

87

92

97

102

Weight (kg)

mean =

sum of scores number of scores

mean =

sum of scores number of scores

mean =

74.5x 3 + 84.5x11 + 94.5x 4 + 104.5x1 19

mean =

77 x 3 + 82 x 5 + 87 x 6 + 92 + 97 x 3 + 102 19

mean =

1635.5 19

mean =

1648 19

Mean = 86.08

Use the midpoint of the group.

Mean = 86.74

Median = 84.5

Median = 87

Range = 104.5 − 74.5 Range = 30

Range = 102 − 77 Range = 25

Use the midpoint of the group.

1 Team individual weights: 99, 85, 80, 83, 87, 88, 96, 92, 75, 88, 76, 103, 87, 83, 83, 86, 84, 79, 95. 2 Daily pollution measurements: 636, 644, 647, 648, 721, 450, 476, 645, 622, 580, 660, 539, 487, 549.

124

No. 3 5 6 1 3 1

Histograms

'Negative skew' The left tail is longer. There are few low values.

'Symmetrical'. mean = median

'Positive skew' The right tail is longer. There are few high values

'Bimodal' Two distinct peaks.

Exercise 9.8 Describe the shape of the following histograms:

The shape is a positive skew.



There are few high values.

1

2

3

4

5

6

Chapter 9 Statistics

125

Comparative Analysis Exercise 9.9 Use a back-to-back stem-and-leaf plot to represent the following data. Use the mean, median, and range to compare the data.

9A Test marks

9B Test marks

68 79 97 78 83 65 72 74 57 63 96 82 67 84

74 72 89 67 78 93 82 79 69 67 60 80 73 79

First, put the data in order: 9A 57, 63, 65, 67, 68, 72, 74, 78, 79, 82, 83, 84, 96, 97 9B 60, 67, 67, 69, 72, 73, 74, 78, 79, 79, 80, 82, 89, 93 9A

sum of scores mean = number of scores 1065 mean = 14

Mean = 76.07 median = (74+78)÷2 Median = 76

9B Test marks (7|8 is 78)

7 8753 9842 432 76

5 6 7 8 9

range = 97 − 57 Range = 40

07799 234899 029 3

mean =

sum of scores number of scores

mean =

1062 14

Mean = 75.86 median = (74+78)÷2 Median = 76 range = 93 − 60 Range = 33

Comment. The stem-and-leaf plot shows that 9A and 9B have a similar slightly positive skew shape. The means of both classes are almost identical and the medians are the same. Thus both classes have no difference on central measures. The range is a little different but not enough to suggest that there is a difference in class performance. Both classes are very similar on this test. 1

2

3

9C Test marks

9D Test marks

78 69 94 73 66 63 71 71 74 65 56 87 59 86

59 61 84 77 83 82 92 86 79 75 73 88 72 74

Fitness measurements of a class immediately after Summer and immediately after Winter.

After Summer

After Winter

69 71 74 53 82 80 66 71 67 50 78 73 60 75 68 43 81

63 45 77 84 56 57 69 70 52 54 67 71 67 71 69 55 47

While researching the strength of concrete beams made from different mixes, the following samples were obtained.

Mix 1 (thousands of psi)

Mix 2 (thousands of psi)

5.4 5.2 6.2 5.0 4.8 5.2 5.2 5.2 6.4 6.0 5.0 6.4 5.8 5.4 5.0 4.8 5.2 5.4 4.8 5.2 5.6 5.6 6.2 5.8 6.0 5.4 5.2 5.8

126

Comparative Analysis Exercise 9.10 Use a compound histogram to represent the following data. Use the mean, median, and range to compare the data.

9C Test marks

9D Test marks

78 69 94 73 66 63 71 71 74 65 56 87 59 86 9C 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99

No. 2 4 5 2 1

59 61 84 77 83 82 92 86 79 75 73 88 72 74

9C

4 2 0

Median = 74.9

No. 1 1 6 5 1

mean =

sum of scores number of scores

mean =

1085 14

6

Frequency

Mean = 72.29

9D

9C 9D

sum of scores mean = number of scores 1012 mean = 14

9D 50 - 59 60 - 69 70 - 79 80 - 89 90 - 99

Group the data.

Mean = 75.86 50-59

60-69

70-79

80-89

Test marks

range = 94.9 − 54.9 Range = 40

90-99

Median = 74.9 range = 94.9 − 54.9 Range = 40

Comment. The histograms show that 9C has a slight positive skew while 9D has a slight negative skew. The mean of 9D is higher than the mean of 9D while the medians and ranges are the same. Thus 9D has a slighly better performance on this assessment than 9C. 1

2

9E Test marks

9F Test marks

78 69 94 73 66 63 71 71 74 65 56 87 59 86

59 61 84 77 83 82 92 86 79 75 73 88 72 74

Weekly earnings of full-time employees:

Female ($)

Male

1250 960 1100 1020 1330 1430 1310 1400 1730 1170 1170 940 930 1650 1090 1580 2010 1330 1340 1490 820 910 1200 1320 710 1070 1310 1600 1460 1740 3 Mean monthly rainfall, mm, for Sydney and Jakarta (capital of Indonesia). Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Sydney 110 122 128 125 123 131 97 86 71 76 85 82 Jakarta 300 300 220 140 130 95 60 45 54 110 140 210 Chapter 9 Statistics

127

Mental Computation

Many everyday problems are solved mentally.

Exercise 9.11 1 Spell Symmetrical. 2 What is the average of 10 and 19? 9 numbers in order. 3 If there are 9 numbers, where is the median? median=(9+1)÷2=5 4 What is the range? The median is the 5th number. 5 What does AAA mean? 6 Scale factor? One side 3, corresponding side 12. 7 Two sides in a right-angled triangle are 1 and 2. Hypotenuse? 2 8 Simplify 18 : 12 2 2 2 9 The car uses 7 litres of petrol for 100 km. h =2 +1 h2=4+1 1 How far will 21 litres take the car? h = √5 10 I bought 3 loaves of bread @ $2.60 each. Cost? 2 + 2 = 5 ???? 2.4 + 2.4 = 4.8 rounding to the nearest integer, 2 + 2 = 5.

Exercise 9.12 1 Spell Histogram. 2 What is the average of 20 and 29? 3 If there are 11 numbers, where is the median? 4 What is the mean? 5 What does SSS mean? 6 Scale factor? One side 5, corresponding side 15. 7 Two sides in a right-angled triangle are 1 and 1. Hypotenuse? 8 Simplify 24 : 18 9 The car uses 8 litres of petrol for 100 km. How far will 20 litres take the car? 10 I bought two 2 litres of milk @ $2.80 each. Cost?

Those who can't laugh at themselves leave the job to others - Anon.

Exercise 9.13 1 Spell Median. 2 What is the average of 30 and 39? 10 numbers in order. 3 If there are 10 numbers, where is the median? median=10÷2=5th and 6th 4 What does bimodal mean? The median is the average of the 5 What does SAS mean? 5th number and the 6th number.. 6 Scale factor? One side 6, corresponding side 18. 7 Two sides in a right-angled triangle are 1 and 3. Hypotenuse? 8 Simplify 30 : 12 9 The car uses 9 litres of petrol for 100 km. How far will 27 litres take the car? 10 I bought 5 loaves of bread @ $2.60 each. Cost?

128

Competition Questions

Prepare for mathematics competitions and build maths muscle at the same time.

Exercise 9.14

Frequency

1

10

If the marks are doubled, draw a histogram to show the new results.

8 6 4 2 20-29 30-39

40-49 50-59

Marks

2

Aaron played a series of soccer matches. His descriptive statistics for the number of goals scored per match is shown. Give an example of a series of matches that would produce these statistics. 1

Aaron

Mean

Median

Range

2

0

6

2

Find the average of 3 and 9

1 1 2 1 3 2 1 5 5 = ( + )= ( + )= ( )= 2 3 9 2 9 9 2 9 18

3

1

1

1

1

5

Find the average of 4 and 2 . 4 Find the average of 4 and 8 . The average of four numbers is 36. If 6 is subtracted from each number, what is then the average of the four numbers?

6

The average of eight numbers is 48. If 2 is subtracted from each number, what is then the average of the eight numbers?

A person has an average of 48 after three tests. What mark must the person get on the fourth test so that the average of the four tests is 50? 48 =

total 3

Total after 3 tests = 144 200 − 144 = 56 Thus need 56 on the fourth test. 7

50 =

total 4

Total after 4 tests = 200

Karen has an average of 83 after ten tests. What mark is needed in the next test to raise her average to 85?

8 Eun-Young has averaged 85% in the six maths test so far this year. Eun-Young would like to finish the year with an overall average of 90% or better. Is this possible with four tests left? 9 The average of five numbers is 4. When the sixth number is added, the average is 5. What is the sixth number? Chapter 9 Statistics

129

Investigations Investigation 9.1 Undertake real-life research. 1 Form a team and brainstorm an appropriate problem or issue. 2

Plan the research • Define the overall research question. • Define subset research questions. • Decide how to obtain data to answer the research questions. • Consider ways of ensuring that the data collection is unbiased. • Consider the equipment needed for the research.

3

Conduct the research • Rehearse the data collection method. • Collect the data.

A backward poet writes inverse.

4

Analyse the data • Look for errors/outliers and decide what to do with the errors/outliers. • Calculate the appropriate descriptive statistics. • Choose an appropriate method of presentation (Histograms etc).

5

Report the conclusions • Match the analysis with the research questions. • Do the answers to the research questions indicate further research questions?

Investigation 9.2 Collect Facebook activity data from a Year 12 or Year 10 form class or maths class and the Facebook activity from your own form class or maths class. Construct compount histograms or stem-and-leaf plots. a) Are the shapes for the other class and your class similar? Would you have expected the shapes to be similar or different? b) Is the data symmetrical? Does the mean of your class have a value close to the value of the median of your class? Does the mean of the other year's class have a value close to the value of the median of the other class? Investigation 9.3 Online statistical activities There are a large number of national and international online statistical activities for Year 9 students. These activities generally involve the collection of data from your class to form a large national data set or international data set. This then forms the basis of infomed research on many relevant topics. Become involved in one of these activities.

Meteorologists study the atmosphere to forecast the weather.

• Relevant school subjects are Mathematics, Science, and English. • Courses usually involve a science degree.

130

A Couple of Puzzles Exercise 9.15 1 Move just one coin so that there are two rows with four coins in each row.

A Game Kayles, described in a 14th century manuscript, is played with a row of skittles or bowling pins. The winner is the person who knocks over the last skittle. (Take it in turns to take one counter or take two counters next to each other). O

O

O

O

O

A strategy can be deduced by working with a smaller number of counters.

O

O

O

O

O

O

O

O

O

O

O

O

O

Play Kayles against the computer. Find a Kayles applet on the Internet.

With three counters, the best first move strategy is to take the middle counter. The other player is then forced to take one from either end, leaving the last counter for you.

A Sweet Trick Use this trick to show off your astonishing memory. 1 Make up a set of cards similar to the following: 1

2 7 189 763

4

3 4 370 774

5 8 640 448

7

6 5 831 459

8 9 101 123

1 561 785

2 022 460 9

6 392 134

3 583 145

2

Ask your audience to choose a card and tell you the card number.

3

You write out the seven-digit number written on the card.

The secret a) Multiply the number of the card by 17 b) Start the number by reversing the digits. c) Add the last two terms to get the third. (Use the last unit only)

4x17 = 68 86 8640448 8+16=14

6+4=10

Chapter 9 Statistics

131

Technology Technology 9.1 The Calculator Scientific calculators will calculate descriptive statistics such as mean and sum. 1 Change the calculator mode to Stat or SD Check your calculator's 2 Clear the calculator. manual for Statistical 2 Enter a number then press M+ calculations 3 Repeat entering a number and then pressing M+ 4 Find the x button, this is the mean. 5

Find the ∑x button, this is the sum of the numbers.

Technology 9.2 The Spreadsheet Most spreadsheets will calculate a massive number of descriptive statistics and draw a frequency polygon: 1 Enter a set of data into the spreadsheet. 2 Enter formulas for the mean, median, and mode. The Internet has 1 2 3 4 5

A 2 2 2 3 4

B C mean = 2.6 median = 2 mode = 2

tutorials for all kinds of spreadsheet graphs.

=mean(A1:A5) =median(A1:A5) =mode(A1:A5)

Technology 9.3 The Graphics calculator A graphics calculator will automatically calculate a large number of descriptive statistics and draw a frequency column graph: 1 2 3

Select the STAT menu, EDIT, and enter data into one of the lists. Return to the main screen. Select the STAT menu, Calc, 1_Stats and enter L1.

To draw a frequency column graph: 1 Enter the numbers in L1 and the frequency in L2. 2 Use STATPLOT to set up the graph Xlist= L1 and Frequency = L2. 3 Use Zoom and ZooMSTAT to fit the graph if necessary. 4 TRACE.

132

L1 2 2 2 3 4

L2

L1 2 3 4

1_Stats x = 2.6 median = 2 n =5 ∑x = 13 + heaps more.

L3

L2 3 1 1

L3

Chapter Review 1 Exercise 9.16 1 Calculate the mean, median, and range for each of the following: a) The ages of the people on the bus: 15, 14, 15, 15, 14, 15, 15, 15, 15, 42 b) Annual Wages: $45k, $52k, $48k, $48k, $53k. 2 Describe the following data distributions using terms such as skew, symmetrical, or bimodal. a) b) Rent (35|5 is 355)

Test marks (6|4 is 64)

5 6 7 8 9

35 36 37 38 39

2 349 23346 447 5

c)

055 0005555 05 5 5

d)

3 Find the mean, median, and range of each of the data displays: a) b) 2 3 3 779 4 122555668

10

Frequency

Test scores (4|5 is 45)

8 6 4 2 74.5

4

84.5

94.5

Weight (kg)

104.5

The skin elasticity of 48 people at a workplace is measured. 24 of the people work mainly outdoors, while the other 24 people work mainly indoors. Analyse the data and make a comment.

Skin elasticity (sun-exposed) 76 56 74 58 82 67 79 55 81 39 65 81 24 45 54 42 38 23 93 42 81 75 86 12

Skin elasticity (sun-protected) 48 88 78 55 44 81 127 67 67 48 31 76 92 105 74 64 80 77 48 59 99 57 70 54 Chapter 9 Statistics

133

Chapter Review 2 Exercise 9.17 1 Calculate the mean, median, and range for each of the following: a) Heart rates: 84, 73, 83, 55, 48, 62, 65, 61, 72 b) Cholesterol levels: 4.4, 5.1, 4.2, 3.8, 4.1. 2 Describe the following data distributions using terms such as skew, symmetrical, or bimodal. a) b) Exports (9|3 is 93)

7 8 9 10 11

Ships (6|2 is 62)

5 244677 35 046689 00

5 6 7 8

c)

0245 2355579 46 2

d)

3 Find the mean, median, and range of each of the data displays: a) b) Test scores (4|5 is 45)

10

Frequency

7 57 8 03588 9 126

8 6 4 2 74.5

4

94.5

104.5

Patients with multiple rib fractures were asked to provide a pain score one hour after receiving one of two analgesic drugs. A high score indicates a high level of pain. Analyse the data and make a comment.

Pain score (Analgesic A) 11 7 7 5 5

134

84.5

Marks

17 7 5 9 7

8 10 11 13 9

5 10 6 3

14 8 12 10

10 4 10 15

Pain score (Analgesic B) 14 13 12 10 8

9 5 10 10 14 5 15 10 5 7 9 10 12 15 13 8 5 15 10 12 5 9

Chapter 6 Proportion When two ratios are equal they are said to be in proportion. Proportion a c If = b d

a b

Inverse proportion means an increase in one quantity will cause a similar decrease in another quantity. Inverse Proportion

c d

a b

then ad = bc or bc = ad

c d

ac = bd

Chapter 7 Pythagoras In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

c

c2 = a2 + b2

a b

The hypotenuse is the longest side. It is opposite the right-angle (90°).

h2=22+12 h2=4+1 h = √5

2 1

Chapter 8 Geometry Similar triangles have exactly the same shape but not necessarily the same size. The corresponding angles are equal. The corresponding sides have the same scale factor. The symbol for similarity is

~.

Two triangles are similar if:

AAA The three matching angles are equal. SSS The three matching sides are in the same ratio. SAS Two matching sides are in the same ratio and the included angles are equal. RHS The hypotenuse and a matching side of a rightangled triangle are in the same ratio.

Chapter 9 Statistics A Stem-and-Leaf Plot is simple, shows the shape of the data and puts the data in order. Test scores (4|5 is 45)

Frequency

10

2 3 3 779 4 122555668

Negative Skew

Histograms: Grouped data on the horizontal axis. Frequencies on the vertical axis. 8 6 4 2 74.5

Symmetrical

Positive Skew

84.5

94.5

Weight (kg)

104.5

Bimodal

135

Review 1 Exercise 10.1 Mental computation 1 Spell Symmetrical. 2 What is the average of 10 and 19? 9 numbers in order. 3 If there are 9 numbers, where is the median? median=(9+1)÷2=5 4 What is the range? The median is the 5th number. 5 What does SSS mean? 6 Scale factor? One side 4, corresponding side 12. 7 Two sides in a right-angled triangle are 1 and 2. Hypotenuse? 2 8 Simplify 12 : 16 9 The car uses 8 litres of petrol for 100 km. h2=22+12 h2=4+1 1 How far will 12 litres take the car? h = √5 10 I bought 5 loaves of bread @ $2.60 each. Cost? Exercise 10.2 1 Simplify the following ratios: a) 5:15 b) 6:10

c) 12:20

d) 15:18

f) 3:1.5

2

e) 1.6:2.4

Which of the following pairs of ratios are in proportion?

a) 6:2 and 15:5

b) 8:2 and 9:3

c) 12:6 and 14:7

d) 2 tonnes are loaded in 5 hours and 6 tonnes are loaded in 15 hours. 3

Draw a graph and write a rule for each of the following:

a) For each 50 km the rally car uses 10 litres of fuel. b) Sound can travel 3.4 km in 10 seconds. 4

Assuming proportionality, solve the following problems.

a) If $AU1 can be exchanged for $US1.04, how many Australian dollars can be exchanged for $US130? b) Riley can run 4 km in 20 minutes. How far can Riley be expected to run in 50 minutes? 5

If 5 people can build a house in 30 days, how long would it take 20 people to build the same house (assuming inverse proportion)?

6

From the graph, find the litres per 100 km fuel consumption for each vehicle. y

a)

Litres (L)

25

b) c)

20 15 10 5

x 0

136

50

100 150 200 250 300

Distance (km)

A mistake is simply another way of doing things Katharine Graham.

7 Which of the following triangles are right-angled triangles? a) b) c) 13

10

8

12

6.3

6

5 8

7.2

3.4

A carpenter needs to know whether a door frame is square. If the door frame is square then a door can be fitted. The door frame measures 810 mm by 2000 mm and the diagonal is 2151 mm.

9 Find the length of the unknown in each of the following: a) 11 b) c) 7.5

15

?

?

9.2

224

330

? 10

A 3.1 m ladder is to be laid against a wall so that the top of the ladder is 2.3m up the wall. How far out from the base of the wall should the ladder be placed? ?

11 Find the length of AB. a) y

b)

y

5

C

4

A

3 2

2

1

1 1 2 3

4 5

-5 -4 -3 -2 -1 0 -1

x

6

-4

c) A(1,2), B(3,5)

1 2 3

4 5

6

x

-2

-2 -3

B

4 3

-5 -4 -3 -2 -1 0 -1

C

5

-3

A -4

B

d) A(4,-1), B(2,5)

12 Use the tests for congruence to test whether the following pair of triangles are congruent:

G

B

9 cm

C

e) A(-3,-2), B(-1,4)

9 cm 12 cm

A

9 cm 12 cm

F 9 cm E

Chapter 10 Review 2

137

13 Enlarge the original triangle by a scale factor of two. a) b) y

y 5

5 4

A

4

C

3

3

A

2

1

1 -5 -4 -3 -2 -1 0 -1

B

C

2

1 2 3

4 5

6

-5 -4 -3 -2 -1 0 -1

x

-2

-2

-3

-3

-4

-4

14a Prove that ABC ~ ADE

1 2 3

4 5

6

x

B

14b Prove that ABC ~ DEC

A

D

B

C C

B

A

E

D

E

15 Find the length of the unknown. a) b)

D

A

8 x

E

D

x

10

C

5 B

E

6

C

18

A

15

B

16 A 1.4 metre stick casts a 0.70 metre shadow at the same time a tree casts a 5.6 m shadow. What is the height of the tree? 17 Calculate the mean, median, and range for each of the following: a) The ages of the people on the bus: 15, 15, 15, 15, 14, 15, 13, 15, 15, 53 b) Annual Wages: $47k, $57k, $54k, $62k, $53k. 18 Describe the following data distributions using terms such as skew, symmetrical, or bimodal. a) b) Rent (35|5 is 355)

35 36 37 38 39

138

055 0005555 05 5 5

19 Find the mean, median, and range of each of the data displays: a) b) 10

Test scores (8|2 is 82)

3 049 12278 3

Frequency

6 7 8 9

8 6 4 2 74.5

84.5

94.5

Weight (kg)

104.5

20 Investigating the use of heat treatment on sprains, a research scientist collected the following data on the time, in days, taken for a muscle sprain to heal. Analyse the data and make a comment.

Sprains (with heat treatment) 15 15 14 20 16 18 15 15 18 18 16 12 18 16 17 16 19 14 19 17 15 12 16 14 18 22 17

Sprains (without treatment) 21 21 20 22 16 25 17 20 19 22 19 20 18 19 22 17 16 20 20 24 21 23 18 22 18 16 19

Review 2 Exercise 10.3 Mental computation 1 Spell Pythagoras. 2 What is the average of 20 and 29? 3 If there are 10 numbers, where is the median? 4 What is the range? 5 What does AAA mean? 6 Scale factor? One side 4, corresponding side 12. 7 Two sides in a right-angled triangle are 1 and 1. Hypotenuse? 8 Simplify 15 : 12 9 The car uses 6 litres of petrol for 100 km. How far will 15 litres take the car? 10 I bought 4 loaves of bread @ $2.60 each. Cost? Exercise 10.4 1 Simplify the following ratios: a) 15:5 b) 12:4

c) 16:8

d) 15:10

f) 5:1.5

2

e) 1.8:1.5

Which of the following pairs of ratios are in proportion?

a) 6:2 and 9:3

b) 5:4 and 10:2

c) 15:5 and 12:4

d) Travel 300 km in 4 hours and travel 450 km in 6 hours. Chapter 10 Review 2

139

3

Draw a graph and write a rule for each of the following:

a) For each 50 km the rally car uses 15 litres of fuel. b) The 200 acre paddock yielded 400 tonnes of corn. 4

Assuming proportionality, solve the following problems.

a) If $AU1 can be exchanged for $US0.90, how many Australian dollars can be exchanged for $US810? b) A car uses 6.5 litres of petrol to travel 100 km. How far will the car travel on 55 litres of petrol? 5

It will cost $55 per person if there are 25 people on the charter bus. How many people would be needed to reduce the cost per person to $50?

6

From the graph, find the litres per 100 km fuel consumption for each vehicle. y

a)

b)

Litres (L)

25

A good head and a good heart are always a formidable combination - Nelson Mandela.

20 15

c)

10 5

x 0

50

100 150 200 250 300

Distance (km)

7 Which of the following triangles are right-angled triangles? a) b) c) 51

45

72

34 63

24 8

3.9

3.6

1.5

A rectangular gate measures 1.4 m by 2.5 m with a 2.79 m diagonal. Is the gate square? If not, should the diagonal be longer or shorter?

9 Find the length of the unknown in each of the following: a) b) c) 9 ?

? 15

4.7

5.6 ?

140

4200

3600

10

A 1.2 m ladder is leaning against a wall. The bottom of the ladder is 0.4 m from the wall and the top of the ladder is 1.13 m up the wall. Is the wall vertical?

11 Find the length of AB. a) y

b)

y 5

5

4

4

A

3

3

2

2 1 -5 -4 -3 -2 -1 0 -1

1 2 3

4 5

-2

C

1

C

6

-5 -4 -3 -2 -1 0 -1

x

c) A(5,2), B(6,5)

-4

d) A(1,-1), B(4,4)

C

e) A(-4,-3), B(4,3) H

12 Use the tests for congruence to test B whether the following pair of triangles are congruent:

50˚

60˚

50˚ A

5

7m

J

y 5

4

4

3

B

2

A

1

A

K

60˚

7m

13 Enlarge the original triangle by a scale factor of two. a) b) y

-5 -4 -3 -2 -1 0 -1

x

6

-3

A

-4

C

4 5

-2

B

-3

B

1 2 3

3 2

C

1 1 2 3

4 5

6

x

-5 -4 -3 -2 -1 0 -1

-2

-2

-3

-3

-4

-4

14 Prove that ABC ~ EBD

4

1 2 3

4 5

6

x

B

Prove that ABE ~ ACD D

C

10

E D 5

A

6

E

3

B

A

B

C

Chapter 10 Review 2

141

15 Find the length of the unknown. A a) b)

D 45

36

D

C 11

9 B

E

8

9

x

C

x

B

A

E

16 A 1.5 metre stick casts a 0.65 metre shadow at the same time a tree casts a 3.6 m shadow. What is the height of the tree? 17 Calculate the mean, median, and range for each of the following: a) Car speed: 56, 45, 54, 55, 53, 62, 49, 51, 72 b) Arm span: 155, 174, 164, 168, 159, 172. 18 Describe the following data distributions using terms such as skew, symmetrical, or bimodal. a) b) Pallets (3|2 is 32)

2 3 4 5

0245 2355699 46 2

19 Find the mean, median, and range of each of the data displays: a) b) 0 7 1 89 2 4556888

10

Frequency

Test scores (1|8 is 18)

8 6 4 2 24.5

34.5

Marks

44.5

55.5

20 Data was collected on daily student absence data during the flu season before and after a handwashing campaign. Analyse the data and make a comment.

Absences (before) 41 46 55 64 57 66 53 57 44 60 47 49 52 41 32 35 33 24 35 20

142

Absences (after) 45 47 52 49 57 52 45 44 43 41 43 31 38 33 36 27 33 23 30 21

Number and Algebra → Real Numbers  Express numbers in scientific notation.  Understand that the use of index notation is an efficient way of representing numbers and symbols and has many applications, particularly in science.  Represent extremely large and small numbers in scientific notation, and numbers expressed in scientific notation as whole numbers or decimals.  Apply index laws to numerical expressions with integer indices.  Apply knowledge of index laws to algebraic terms and simplify algebraic expressions, using both positive and negative integral indices. Measurement and Geometry → Units of Measurement  Investigate very small and very large time scales and intervals.  Investigate the usefulness of scientific notation in representing very large and very small numbers.

A TASK I'm off to compete in the Mental Calculation World Cup.

Design, develop, and evaluate a survey to test the support for the following statement: "Mental computation is so important that it should be taught and tested as a part of every maths lesson".

A LITTLE BIT OF HISTORY Zerah Colburn (1804-1840) showed awesome mental powers at the age of just eight years old. Zera could instantly give the product of two numbers each of four digits but hesitated if both numbers exceeded 10 000.

Zera was thought to be mentally retarded until seven.

Zera was asked for 816 in a few seconds, he replied '281 474 976 710 656'. Asked for the factors of 247 483 he replied 941 and 263. Asked for the factors of 171 395 he gave 5, 7, 59 and 83. Asked for the factors of 36 083 he said there were none.

143

Warmup A convenient way of writing 2×2×2 is

2

3

2 8×8×8×8 5 b×b×b×b×b 8 10×10×10×10

Base Indices save a lot of effort.

Exercise 11.1 Write the following in index form: 2×2×2×2×2 a×a×a×a = 25 = a4 1 2×2×2 4 10×10×10 7 m×m×m×m×m

Index

3 a×a×a×a×a 6 h×h×h 9 3×3×3×3×3×3

Exercise 11.2 Write the following in factor form: 104 b3 = 10×10×10×10 = b×b×b = 10 000 1 43 4 27 7 x4

2 b4 5 62 8 p5

3 52 6 m5 9 14

Exercise 11.3 Write the following in index form: 2×2×2×4×4 abbaaab 3 2 = 2 ×4 = a4×b3 1 aabbbaa 2 3×3×3×2×2 3 abaaababb 4 2×10×2×10×2×10×2 5 bggggbbbg 6 zzzzzzzzzzz 7 ppqrppqqrrrp 8 2×2×2×3×4×4×4×3 9 10gg10g10g10gg Exercise 11.4 Simplify the following: 104×102 = 10×10×10×10×10×10 = 106 1 a2×a3 4 103×103 7 (3×102)3

144

(2×102)3 = 2×102×2×102×2×102 = 23×106 2 33×34 5 b5×b 8 (a×102)3

3 a4×a2 6 102×103×104 9 (32×103)2

Index Law 1 Multiplying Indices:

Multiplying Indices: 24×22

or

= 2×2×2×2 × 2×2 = 26

24×22

Index Law 1 am×an = am+n

= 24+2 = 26

Exercise 11.5 Simplify and write the following in index form: 103×102 = 10×10×10 × 10×10 = 105 a2×a5 = a×a × a×a×a×a×a = a7 or 103×102 = 103+2 = 105 1 102×103 5 x2×x4 9 23×21 13 b4×b3 17 6.35×6.32 21 104×103×102

or a2×a5 = a2+5 = a7

2 42×42 6 x3×x2 10 a2×a7 14 102×107 18 e3×e6 22 z4×z3×z2

3 34×33 7 x5×x2 11 32×33 15 c3×c3 19 0.15×0.14 23 22×23×23×24 b = b1 10 = 101

Index Law 2 Dividing Indices: a3÷a2 =

a ×a ×a a ×a

4 104×103 8 x4×x3 12 103×106 16 d×d4 20 105×103 24 y2×y2×y3

=a

Index Law 2

Dividing Indices:

or

a3÷a2

am÷an = am−n

= a3−2 = a

Exercise 11.6 Simplify and write the following in index form: 103÷102 =

10 ×10 ×10 10 ×10

a ×a ×a ×a ×a ×a a ×a

= 10

a6÷a2 =

or 103÷102 = 103−2 = 10

or a6÷a2

1 105÷103 5 x5÷x2 9 103÷10 13 y7÷y3 105

19

a ×a a4

= a4

3 26÷22 7 107÷103 11 26÷24 15 109×105÷106

4 35÷33 8 b4÷b3 12 106÷102 16 x3÷x3

x7

17 103 7

2 104÷102 6 a4÷a2 10 n8÷n4 14 0.24÷0.22

= a6−2

= a×a×a×a = a4

18 x 4 2



7

m4÷m2 = 3

10 ×10

20 104 ×106

m4 m2

They are the same thing.

Chapter 11 Indices 2

145

Index Law 3 Power Indices: (23)2 = (2×2×2)2 = (2×2×2)×(2×2×2) = 26

Power Indices: (23)2 = 23×2 = 26

or

Index Law 3 (am)n = am×n

Exercise 11.7 Simplify and write the following in index form: (b4)2 = (b×b×b×b)2 104×(102)3 = 104×106 = 1010 = (b×b×b×b)×(b×b×b×b) = b8 (b4)2b3 = b8×b3 = b11 or (b4)2 = b4×2 = b8 1 (b2)3 5 (102)2 9 (102)2 13 (43)3 17 (103)3 21 102×(102)3 25 103(102)2 29 (32)2×3

2 (b3)2 6 (a3)4 10 (x2)5 14 (102)3 18 (c4)3 22 22×(23)3 26 b5(b3)4 30 p2(p2)3

3 (b3)3 7 (32)4 11 (y2)3 15 (10)3 19 (d2)5 23 (x2)3×x3 27 (22)425 31 10(102)3

4 (102)3 8 (22)4 12 (55)2 16 (a2)5 20 (0.35)3 24 z5×(z2)3 28 m3(m3)5 32 10×(104)2 10 = 101

The more practice you get the easier it becomes.

Index Law 4 Zero Index: p3÷p3 = 1 or p ÷p = p = p0 Which must be = 1 3

3

3−3

or

Exercise 11.8 Simplify each of the following:

146

100 = 1

h0 = 1

1 100 5 b0 9 3×100 13 8m0 17 (20)2×2

2 20 6 30 10 4c0 14 2h0 18 p2(p0)3

Zero Index

Zero Index:

a0 = 1

p0 = 1

Try 50 on your calculator. Is your answer 1?

3×50 = 3×1 = 3 3 x0 7 40 11 9×20 15 5×100 19 10(104)0

5b0 = 5×1 = 5 4 a0 8 y0 12 3×10 16 6×0.010 20 10×(100)3

Index Law 5 What happens when the index is negative?

or a2÷a5 =

a ×a 1 = a ×a ×a ×a ×a a ×a ×a

Negative Index

a ÷a 2

=a

5

2−5

a−m =

−3

=a

1 am

Exercise 11.9 Write each of the following using a negative index: 1

= 10−3 10 3

1

1

10 2

1

5

x

9

1 100

3



1

1 10

= b−5 b 5

1 10000

= 10−1

1

2

5

10

3

6

1 10

7

1

10 1000

1

= 104 = 10−4

1

4

b 4

1

8

10 4

1

1 35 1 26 1

11 8

12 27

Simplify and write the following in index form: 102×10-3 = 102-3 = 10-1 10−3 ÷10−4 = 10-3- -4 = 10-3+4 = 10 13 10-2×103

14 10-2×105

15 106÷10-2

16 10-2÷10-2

17 3-4×33

18 10-4×105

19 x5÷x-4

20 10-2÷105

21 x6×x-4

22 103×10-4×102

23 26÷22

24 35÷33

25 x-5×x2×x4

26 a4×a-9×a3

27 10-7÷103

28 b-4÷b-3

(10−2)4 = 10ˉ2×4 = 10−8

9(100)-3 = 9×100×-3 = 9×1 = 9

29 (10−2)3 33 (x2)3

30 (5−3)3 34 (10−3)2

31 (22)−4 35 (b4)−3

32 (10−2)−2 36 (y−1)−6

37 3×100

38 4(c-2)0

39 9(100)-3

40 3×10

4×105×2×10-3 = 8×105-3 = 8×102

6×10-2÷(2×10-5) = 3×10-2- -5 = 3×10-2+5 = 3×103

41 3×102×2×103

42 6×106×3×10-3

43 9×10-2÷(3×10-6) 44 4×103÷(2×10-4)

45 4×104×3×105

46 2×10-4×3×10-2 47 x5×4x−3÷2x2

48 (x−2)2×6x3÷2x2

Chapter 11 Indices 2

147

Scientific Notation

Very large numbers. The Sun has a mass of: 2 000 000 000 000 000 000 000 000 000 000 kg. 2×1030 kg

Index notation is easier and quicker.

6

Exercise 11.10 Write in scientific notation: 314 000 = 3.14×105

4 670 000

5 4 3 2 1

4 670 000

= 4.67×106

1

375 000

2

582 000

3 160 000

4

5 100 000

5

6 200 000

6 7 400 000

7

47 000

8

29 000

9 81 300

10 83 000 000

11 48 100 000

12 7 620 000

13 91 000

14 76 000 000

15 20 000 000

16 The power station produces 250 000 000 000 watts. 17 The speed of light, 299 800 000 m/s. 18 The distance from the Earth to the Sun, 149 000 000 000 m. 19 The diameter of the Earth, 12 700 000 m 20 Estimated number of stars in the Milky Way, 100 000 000 000 stars. Exercise 11.11 Write as ordinary numbers: 5.7×104 = 57 000

1

9.13×107 = 91 300 000

91 300 000

1 103

2 3×105

3 8×104

4 4×105

5 3.3×102

6 8.1×108

7 6.3×107

8 7.11×107

9 109

10 3.879×105

11 9.324×106

12 7.8214×109

13 2.1012×1012

14 5.5×108

15 2.3×104

16 The amount of water in the dam, 3.2×108 litres.

148

2 3 4 5 6 7

Scientific Notation

Very small numbers. Electric charge of an electron: 0.000 000 000 000 000 000 160 2 coulombs. 1.602×10-19 coulombs

Index notation is easier and quicker.

1

Exercise 11.12 Write in scientific notation:

2 3 4

5

0.000 081

= 0.000 174

= 0.000 081

= 1.74×10-4

= 8.1×10-5

0.001 =

1 1 = 3 = 10−3 1000 10

1

0.000 031

2

0.000 006 3

3 0.001

4

0.000 8

5

0.000 856

6 7 400 000

7 0.05

8 0.26

9 0.491

10 0.001

11 0.000 1

12 0.000 01

13 0.009 1

14 0.000 066

15 0.000 000 0939

16 A virus has a length of: 0.000 000 025 m. 17 Light travels one metre in: 0.000 000 003 s. 18 Wavelength of green light: 0.000 000 52 m. 19 Mass of an electron: 0.000 000 000 000 000 000 000 000 000 9 g 20 Mass of a proton: 0.000 000 000 000 000 000 000 001 67 g Exercise 11.13 Write as ordinary numbers: = 2×10−4 = 0.000 2

1

= 4.5×10−8 = 0.000 000 045

2 3 4 5 6

7 8

0.000 000 045

1 2×10-1

2 7×10-2

3 5×10-3

4 1×10-4

5 6.7×10-5

6 3.2×10-6

7 8.35×10-4

8 9.91×10-9

9 1.04×10-8

10 6.172×10-6

11 4.304×10-7

12 6.826×10-12

13 9.013×10-9

14 6.5×10-12

15 3.3×10-15

16 The muon has a mean lifetime of 2.2×10-6 seconds. Chapter 11 Indices 2

149

Scientific Notation Exercise 11.14 Light travels 3×108 metres in one second. How far will light travel in one minute?

See Technology 11 for calculator use.

3×108 m in 1s x m in 60s

x×1 = 3×108×60 {cross multiply} 8 x = 180×10 Using a calculator: x = 1.8×1010 metres 3 EXP 8 × 60 =

1

Light travels 3×108 metres in one second. a) How far will light travel in one minute? b) How far will light travel in one hour? c) How far will light travel in one day?

2

A light year is used to describe distances in space. A light year is the distance light will travel in one year. Light travels 3×108 m/s (ie., 3×108 metres in one second). How far does light travel in a light year?

3

The Moon is approximately 3.8×108 m from the Earth. How long will it take to get from the Earth to the Moon at a speed of 1800 m/s (ie 1800 metres in 1 second)?

4

Alpha Centauri, our nearest star, is approximately 4.3 light years away from the Earth. How long will it take to a spacecraft to get from the Earth to Alpha Centauri at a speed of 100 000 km/h?

Light travels 3×108 metres in one second. How long does it take light to travel 1 metre? 3×108 m in 1s 1 m in xs

x×3×108 = 1×1 1

{cross multiply}

x = 3×108 x = 0.000 000 003 metres = 3×10-9 s

Using a calculator: 1

÷

5

Light travels 3×108 metres in one second. How long does it take light to travel 100 metres?

6

Light travels 3×108 metres in one second. How long does it take light to travel 1 km?

3 EXP 8

=

7 The circumference of the Earth is approximately 4×104 km and the Earth rotates once every 24 hours. What is the speed of the surface

150

of the Earth in metres per second ( speed =

dis tan ce time

)?

Scientific Notation Exercise 11.15 Find the circumference of the Earth (Radius = 6.38×106 m). C = 2πr C = 2π×6.38×106 C = 40086722.26 m C = 4×107 m

Using a calculator: 2

× π × 6.38 EXP 6

=

1

Find the circumference of the Moon (Radius = 1.74×106 m).

2

Find the circumference of the Sun (Radius = 6.95×108 m).

3

Find the circumference of a classical electron orbit (Radius = 2.82×10-15 m).

Find the volume of the Earth (Radius = 6.38×106 m). V=

4πr 3 3

4π(6.38 ×106 )3 V= 3

Using a calculator: 4

× π ×

( 6.38 EXP 6

) yx 3

=

V = 1.09×1021 m3 4

Find the volume of the Moon (Radius = 1.74×106 m).

5

Find the volume of Mars (Radius = 3.38×106 m).

6

Find the volume of the Sun (Radius = 6.95×108 m).

7

Are the Earth, Moon, and Mars composed of similar material? a) Find the density of Earth (Mass = 5.975×1024 kg). mass Density = b) Find the density of Moon (Mass = 7.35×1022 kg). volume c) Find the density of Mars (Mass = 6.42×1023 kg).

Our Universe 1.4×1010 years old? 1.3×1011 galaxies? 5×1022 stars? 9 × 1026 metres in width?

Chapter 11 Indices 2

151

Mental Computation Exercise 11.16 1 Spell Scientific. 2 103×104 3 Write in scientific notation: 570 000 4 Write in scientific notation: 0.000 49 5 What is the average of 2, 2, 2, 4, 6? 6 What is the median of 2, 2, 2, 4, 6? 7 What does SSS mean? 8 Two sides in a right-angled triangle are 1 and 1. Hypotenuse? 9 Simplify 15 : 9 10 The car travels 50 km in 40 s. At this speed, how long will it take the car to travel 150 km?

Exercise 11.17 1 Spell Notation. 2 10-3×104 3 Write in scientific notation: 2 340 000 4 Write in scientific notation: 0.000 023 5 What is the average of 2, 2, 3, 4, 5? 6 What is the median of 2, 2, 3, 4, 5? 7 What does SAS mean? 8 Two sides in a right-angled triangle are 1 and 2. Hypotenuse? 9 Simplify 16 : 12 10 The car travels 50 km in 50 s. At this speed, how long will it take the car to travel 200 km?

Mental computation helps people prepare for problems in everyday life.

1 1 h2=12+12 h2=1+1 h = √2

Even if you fall on your face, you're still moving forward - Victor Kiam.

Incoming fire has the right of way - Murphy's Laws of Combat.

Exercise 11.18 1 Spell Measurement. 2 10-3÷104 3 Write in scientific notation: 5 540 000 000 4 Write in scientific notation: 0.000 000 6 5 What is the average of 2, 2, 3, 3, 6? 6 What is the median of 2, 2, 3, 3, 6? 7 What does AAA mean? 8 Two sides in a right-angled triangle are 3 and 1. Hypotenuse? 9 Simplify 25 : 15 10 The car travels 60 km in 40 s. At this speed, how long will it take the car to travel 180 km?

152

Competition Questions Build maths muscle and prepare

Exercise 11.19 for mathematics competitions at 1 Find the value of each of the following: the same time. 2 a) (0.1) b) (0.01)2 c) (0.001)3 (0.01)2 = (10-2)2 = 10-4 or = 0.0001 {4 decimal places} d) (0.0001)2 e) 10×102×103 f) 10×102×103×104×105 g) 2×22×23×24×25 h) 1 + 4/100 + 6/10000 2

Which is the largest? a) 3×32×33

b) (3×32)3

c) 32×3÷3-2

3 Light travels 3×108 metres in one second. How far will light travel in 1 minute? What is the last digit in: 312×415? 31 = 3 32 = 9 33 = 27 34 = 81 35 = 243 Last digit pattern is: 3, 9, 7, 1, 3, 9, ... Last digit in 312 is 1

41 = 4 42 = 16 43 = 64 44 = 246 Last digit pattern is: 4, 6, 4, 6, 4, .... Last digit in 415 is 4 Last digit in 312×415 is 1×4 = 4

4

What is the last digit in: 313×416?

5

What is the last digit in: 216×443?

6

What is the last digit in: 519×413?

7

What is the last two digits in: 5555?

8

Given that a and b can be any of 2, 3, 4, and 5,



what is the largest possible value of

(b − a)(b−a)?

Ship's Officers operate as an intermediary between the captain and the crew. They supervise the deck crew as they load and unload cargo. • Relevant school subjects are English, Mathematics, Physics. • Courses normally involve a cadetship with a shipping company.

Chapter 11 Indices 2

153

Investigations Investigation 11.1

1 million What is the height, in metres, of a million dollars worth of $50 notes if each $50 note is 0.25 mm thick?

What is the height, in metres, of a million sheets of paper 0.25 mm thick?

What is the height, in metres, of a million dollars worth of $10 notes if each $10 note is 0.25 mm thick?

Investigation 11.2 A micrometre is one millionth of a metre.

How far will a million steps take you?

Micrometre

Investigate Ways in which people can understand the size of a micrometre?

centimetre = 0.01 m millimetre = 0.001 m micrometre = 0.000 001 m nanometre = 0.000 000 001 m picometre = 0.000 000 000 001 m

Investigation 11.2 10 1015 1012 109 106 103 102 10 10-1 10-2 10-3 10-6 10-9 10-12 10-15 10-18 18

154

Prefix exa peta tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto

Scientific Notation

Investigate Practical applications of very large numbers.

Investigate Practical applications of very small numbers.

A Couple of Puzzles Exercise 11.20 1 If you lived 1 billion seconds, how old would you be? 2

1 billion (common) = 1 thousand million = 109 = 1 000 000 000

1 billion (old meaning) = 1 million million = 1012 = 1 000 000 000 000

Sal, Mal, and Al each belong to a different club. One is in the maths club. One is in the chess club. One is in the science club. The chess club member is the youngest. Mal is older than the maths club member. The maths club member's brother would like to join the maths club. Sal is an only child. Name the person in each club Maths

Chess

Science

Sal Mal Al

A Game Alphabet is played by two people with the 26 letters of the alphabet. The winner is the person who crosses out Z. A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Starting at A, take it in turns to cross out 1, 2, 3 or 4 letters. The letters must be in order.

Play a couple of games and try to determine a winning strategy.

A Sweet Trick 1 2 3 4 5 6

Ask your audience to choose a number from 1 to 10. Add 10. Double. Subtract 4. Double. Divide by 4.

6 6+10 = 16 2×16 = 32 32−4 = 28 2×28 = 56 56÷4 = 14 An algebraic explanation:

Ask for their answer and subtract 8 to get their number. 14 − 8 = 6

=

2(2( x + 10) − 4) 4

=

2(2 x + 20 − 4) 4 4 x + 32

= 4 =x+8

Chapter 11 Indices 2

155

Technology Technology 11.1 Calculators and Indices 23×32 =

2 yx 3 × 3

The power button can be:

yx 2 =

yx

or

^

= 72 Technology 11.2 Calculators and Scientific Notation

Light travels 3×108 metres in one second. How far will light travel in one day?

3×108 m x m

in in

1s 24×60×60 s

x×1 = 3×108×24×60×60 3 EXP 8 × 24

× 60

{1 day = 24×60×60 s } {cross multiply} × 60 =



x = 2.59×1013 Light travels 2.59×1013 metres in one day.



a) How far will light travel in one minute? b) How far will light travel in one hour?

Technology 11.3 Calculators and Scientific Notation

Find the volume of the Earth (Radius = 6.38×106 m). 4πr 3

V= 3

V=

4π(6.38 ×106 )3 3

4 × π ×

( 6.38 2nd EE 6

)

^

3 ÷

V = 1.09×1021 m3 The volume of the Earth is 1.09×1021 cubic metres. a) Find the volume of the Moon (Radius = 1.74x106 m). b) Find the volume of Mars (Radius = 3.38x106 m). c) Find the volume of the Sun (Radius = 6.95×108 m).

156

3

Enter

Chapter Review 1 Exercise 11.21 1 Write each of the following using a negative index: 1

a) 103 2

1

b) 104

c)

1 a7

Simplify and write the following in index form:

a) 10-3×105

b) 10-2×104

c) 103÷10-7

d) 10-4×10-3

e) 102×10-4×106

f) x4×x-8×x3

g) (10−2)3

h) (2−3)3

i) (52)−3

j) (10−2)−3

k) (a2)3

l) 8×10

m) 5×102×3×103

n) 2×106×3×10-4

o) 6×10-4÷(3×10-6)

p) 3×10-4×5×10-3

q) x6×4x−3÷2x2

r) (x−2)2×8x4÷2x2



3

Write in scientific notation:

a) 95 000

b) 520 000

c) 16 000 000

d) 0.000 01

e) 0.000 005 2

f) 0.005

g) 12 grams of carbon has 60 000 000 000 000 000 000 000 molecules. h) The virus has a length of: 0.000 000 04 m. 4

Write as ordinary numbers:

a) 102

b) 6.2×107

c) 9.51×108

d) 2×10-3

e) 7.42×10-4

f) 4.55×10-6

g) The population of Indonesia is approximately 2.3×108. h) Mass of an electron: 9×10-28 g 5

Find the circumference of Mars (Radius = 3.4×106 m). Circumference (Distance around the outside.) C=2πr

6

Find the volume of Mars (Radius = 3.4×106 m). Volume (Space occupied by the sphere.) V=

7

4πr 3 3

What is the last digit in: 313×416? Chapter 11 Indices 2

157

Chapter Review 2 Exercise 11.22 1 Write each of the following using a negative index: 1

a) 102 2

1

b) 105

c)

1 x9

Simplify and write the following in index form:

a) 10-2×103

b) 10-5×103

c) 106÷10-2

d) 10-5×10-3

e) 103×10-4×105

f) x5×x-7×x2

g) (10−3)3

h) (5−3)4

i) (22)−4

j) (10−3)−3

k) (y2)5

l) 9×10

m) 2×104×3×103

n) 5×107×3×10-3

o) 4×10-4÷(2×10-5)

p) 2×10-3×4×10-3

q) a6×4a−4÷2a2

r) (x−2)3×8x4÷2x5



3

Write in scientific notation:

a) 430 000

b) 23 000

c) 200 000 000

d) 0.001

e) 0.000 072

f) 0.000 09

g) Jupiter has a diameter of 140 000 000 m.. h) The bacteria has a length of: 0.000 006 m. 4

Write as ordinary numbers:

a) 104

b) 5.5×106

c) 8.01×109

d) 4×10-3

e) 1.42×10-6

f) 3.05×10-12

g) The population of Australia is approximately 2.3×107. h) The average size of a nucleus in an animal cell is 9×10-7 metres. 5

Find the circumference of Jupiter (Radius = 7.0×107 m). Circumference (Distance around the outside.) C=2πr

6

Find the volume of Jupiter (Radius = 7.0×107 m). Volume (Space occupied by the sphere.) V=

7

158

What is the last digit in: 316×416?

4πr 3 3

Measurement and Geometry  Pythagoras and Trigonometry  Use similarity to investigate the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles.  Develop an understanding of the relationship between the corresponding sides of similar right-angled triangles.  Apply trigonometry to solve right-angled triangle problems.  Understand the terms ‘adjacent’ and ‘opposite’ sides in a right-angled triangle.

And with a torch it even works at night.

A TASK A sundial measures the time using the position of the sun. Use the Internet to find a sundial design (there are hundreds of different designs). • Make your sundial. • Test your sundial. • Demonstrate your sundial to your class.

A LITTLE BIT OF HISTORY Around 2 000 years ago, Indian astronomers developed trigonometry based on a sine function. The Indian sine function was the length of the opposite side for a given hypotenuse. Muslim scientists had tables for sine and tangent that were extremely accurate (1 part in 700 million). When calculus was invented, around 300 years ago, trigonometric functions became much more important in many more pure and applied mathematical applications.

e

us

en ot

p

hy

A

α

opposite

The Babylonians, around 3 000 years ago, measured angles in degrees, minutes, and seconds.

adjacent

sin A =

opposite hypotenuse

adjacent hypotenuse opposite tan A = adjacent

cos A =

159

Pythagoras' Theorem c2 = a2 + b2

In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

c

a b

The hypotenuse is the longest side. It is opposite the right-angle (90°). Exercise 12.1 Find the length of the hypotenuse in each of the following: First add a, b, c c2 = a2 + b2 2 c=? c = 532 + 472 ? 53 a=53 2 c = 5018 c = 5018 47 b=47 c = 70.84 1 2 3 15

? 7

?

35 32

66

4 5 6 ? 2.6 ? 23 38

10 m

7 A 3m wide by 1.4 m rectangular gate needs a diagonal brace to keep it rigid. What should be the length of the diagonal? A builder checks the right-angle of a slab corner by making marks 10 m out from each corner. How far apart should the marks be?

? 10 m

160

86.2 73.5 ?

3.1

8

?

87

Pythagoras' Theorem In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

a2 + b2 = c2

c

a b

Exercise 12.2 Find the length of the unknown in each of the following: First add a, b, c a2 + b2 = c2 2 6.4 c=6.4 a + 5.12 = 6.42 a2 = 6.42 − 5.12 2 b=5.1 a = 14.95 5.1 ? a=? a = 14.95 a = 3.87 1 2 3 6.2

7.8

4.6

?

5.3

?

743

1509

?

Rounding to two decimal places, first look at the third decimal place: 56.231694 27.01769 1.07276 4.79634216 less than 5 thus 56.23 5 or more thus 27.02

less than 5 thus 1.07

4

A 45 m tower is supported by guy wires. The guy wires are attached to the top of the tower and anchored to the ground out from the tower. If the guy wires are 150 m long, how far out from the tower are they anchored?

5

What is the distance from A to B on the 10 cm cube?

5 or more thus 4.80

B

A Chapter 12 Trigonometry 1

161

Naming Sides The hypotenuse is the longest side. It is opposite the right-angle.

Opposite is 'opposite' the angle.

θ

opposite

po

hy

e

us

ten

adjacent

Adjacent means 'next to' the angle.

Exercise 12.3 For each of the following triangles, name a) the hypotenuse. b) the side adjacent to the angle. c) the side opposite the angle.

A

AC is the hypotenuse. BC is adjacent. AB is opposite. α

B

C

C

β

A AB is the hypotenuse. AC is adjacent. BC is opposite.

B

1 2 3 C C

C

β α

A

B

φ A

B

A

B

4 5 6 θ

B

A

A B

C

162

C α

λ C

A

B

Trigonometry E

We have made use of the ratios of corresponding sides of similar right-angled triangles for thousands of years.

F

G

A

α C

B

D

You will need a ruler to measure each line.

Exercise 12.4 1 Copy and complete the following table: Triangle ABG ACF ADE

2

Adjacent

1.7 cm

4.6 cm

opposite adjacent

tan A =

1.7 ÷ 4.6 = 0.37

opposite adjacent

α = 20° Use your calculator:

tan 20 =

Complete the following table: Triangle

ABG ACF ADE

3

Opposite

Opposite 1.7 cm

Hypotenuse 4.9 cm

opposite hypotenuse

1.7÷4.9 = 0.35

sin A =

opposite hypotenuse

α = 20° Use your calculator:

sin 20 =

Complete the following table: Triangle

ABG ACF ADE

Adjacent 4.6 cm

Hypotenuse 4.9 cm

adjacent hypotenuse

4.6÷4.9 = 0.94

cos A =

adjacent hypotenuse

α = 20° Use your calculator:

cos 20 =

Chapter 12 Trigonometry 1

163

The Tan Ratio Trigonometry was developed thousands of years ago to solve the many problems in surveying, engineering, architecture, astronomy, etc, etc, etc.

opposite adjacent

e

us

en ot

opposite

tan θ =

Trigonometry n. branch of mathematics dealing with the relationships between angles and sides of triangles.

p

hy

θ

adjacent

Exercise 12.5 Find tan α and the size of the angle α. 3 is opposite α



5

3 4 is adjacent to α



4

α

tan α =

opposite adjacent

tan α =

3 4

= 0.75

α = tan-1 0.75 α = 36.9°

Use your calculator:

2ndF tan-1 0.75 =

tan-1 means 'an angle whose tan is'. Thus tan-1 0.75 means 'an angle whose tan is 0.75' (which is 36.9°).

Make sure your calculator is on degrees.

1 2 3 5 α

3 4

10 α

6

8

4 5 6 5 17 α 12 α 15 13 8

164

50 α

30

40

5

12 13

α

The Tan Ratio Trigonometry can be used to find a side after knowing a side and an angle in a right-angled triangle.

Trigonometry is used millions and millions of times every day.

Exercise 12.6 Find x in each of the following right-angled triangles:

opposite adjacent x tan 23 = 47 tan α =

x 23° 47

tan 23×47 = x 19.95 = x

{inverse of ÷ is ×}

Make sure your calculator is in degrees (deg).

tan 23 × 47 =

1 2 3 x

x

x

34° 12

29°

41°

68

7.5

4 5 6 1.4 35° x 55° 7.5 x 7

20 m out from the base of a tree, a clinometer measures the angle of elevation to the top of the tree as 54°. Find the height of the tree.

8

15 m out from the base of a flagpole, a clinometer measures the angle of elevation to the top of the flagpole as 39°. Find the height of the flagpole.

39°

x

9.2

x 54° 20

x 15

39°

Chapter 12 Trigonometry 1

165

The Tan Ratio Trigonometry can be used to find a side after knowing a side and an angle in a right-angled triangle.

Pythagoras' Theorem can be used to find the third side after knowing two sides in a right-angled triangle.

In any right-angled triangle: The square on the hypotenuse is equal to the sum of the squares on the other two sides.

Exercise 12.7 Find the unknown sides: 60°

y

60°

c=y

5.7

x

b = 5.7

c2 = a2 + b2

a = 9.9

opposite adjacent x tan 60 = 5.7 tan α =

tan 60×5.7 = x 9.87 = x

c2 = a2 + b2 y2 = 9.92 + 5.72 y2 = 130.5 y = √130.5 y = 11.40

c

a b

1 2 3 y

x

y

x

58° 2.9

9.9

y 39°

x

54° 7.4

4 5 6 y x

x 76° 3.2

y

7.03

x 43°

y

19.6

27°

7

A student with a clinometer, is lying on the ground 4.6 m out from the base of a flagpole. If the clinometer reads 45°, what is the height of the flagpole?

8

6.2 m out from the base of a tree, a clinometer measures the angle of elevation to the top of the tree as 34°. Find the height of the tree.

9

4.3 m out from the base of a building, a clinometer measures the angle of elevation to the top of the building as 45°. Find the height of the building.

10 The angle of elevation of the top of a tower from a point 37 m out from the base of the tower is 53°. Find the height of the tower correct to one decimal place.

166

The Tan Ratio The angles in a triangle sum to 180°

Trigonometry can be used to find an angle after knowing two sides in a right-angled triangle.

Pythagoras' Theorem can be used to find the third side after knowing two sides in a right-angled triangle. Solve means 'find all unknowns'. A triangle has 3 sides and 3 angles.

Exercise 12.8 Solve the following triangles: β

c

3.1

a = 3.6

opposite adjacent 3.1 tan α = 3.6 tan α =

c2 = a2 + b2 c2 = 3.62 + 3.12 c2 = 22.57 c = √22.57 c = 4.75

α = tan-1(3.1÷3.6) α = 40.7°

3.1

40.7° 3.6

α 3.6

49.3°

4.7

b = 3.1

sum of angles = 180° α + β + 90 = 180 40.7 + β + 90 = 180 β = 180 − 40.7 − 90 β = 49.3°

1 2 3 β

c

4.7

α

c

7.8

α 8.3

50

α

β

β

9.9

c

50

4 5 6 19.6

α

30

c

40.3

α

β

c 52

35

β

c β 44

tan θ =

The tangent ratio is one of several ratios involving the relationships between the sides and angles of triangles. Sin and cos are in Chapter 17.

α

sin A =

opposite hypotenuse

opposite adjacent cos A =

adjacent hypotenuse

Chapter 12 Trigonometry 1

167

Mental Computation

Mental computation gives you

practice in thinking. Exercise 12.9 1 Spell Trigonometry. 2 What is the tan ratio? 3 In the triangle, what is tanα? 4 If one angle in a right-angled triangle is 30°, 3 what is the third angle? α 5 Two sides in a right-angled triangle are 1 and 3. 1 Hypotenuse? c2=12+32 6 Write in scientific notation: 54 000 c2=1+9 7 Write in scientific notation: 0.003 2 c = √10 8 106÷104 9 What is the average of 2, 2, 3, 4, 5? 10 16×25 16×25 = 4×4×25 = 4×100 = 400

Exercise 12.10 1 Spell Tangent. 2 What is the tan ratio? 3 In the triangle, what is tanα? 4 If one angle in a right-angled triangle is 60°, what is the third angle? 5 Two sides in a right-angled triangle are 2 and 3. Hypotenuse? 6 Write in scientific notation: 170 000 7 Write in scientific notation: 0.000 14 8 106÷103 9 What is the average of 1, 2, 3, 4, 5? 10 20×25 Exercise 12.11 1 Spell Pythagoras. 2 What is the tan ratio? 3 In the triangle, what is tanα? 4 If one angle in a right-angled triangle is 40°, what is the third angle? 5 Two sides in a right-angled triangle are 2 and 2. Hypotenuse? 6 Write in scientific notation: 3 000 000 7 Write in scientific notation: 0.000 000 9 8 109÷106 9 What is the average of 2, 3, 3, 4? 10 24×25

If you can dream it, you can do it - Walt Disney

3 α

2

All of us could take a lesson from the weather. It pays no attention to criticism..

2 α

2

Conveyancers compile the documentation needed for the sale and purchase of real estate. • Relevant school subjects are English and Mathematics. • Courses usually involve a diploma or business degree.

168

Competition Questions Build maths muscle and

Exercise 12.12 prepare for mathematics 1 What is the square root of 400? competitions at the same time. 2 What is the square root of 4? √0.0009 = 0.03 3 What is the square root of 0.04? 4 What is the square root of 0.0004? gradient=slope=tangent ratio. 5 What is the gradient of the ramp, Assume each block is square. the thick line, in each of the following? a) b) c)

6 7 8

One angle in a right-angled triangle is 37°, what is the size of the other two angles? Two sides of a triangle are 6 cm and 3 cm. Can the third be 10 cm? Two sides of a triangle are 6 cm and 3 cm. Can the third be 2 cm?

A right-angled isosceles triangle has an area of 18. What is the length of the hypotenuse? c2 = a2 + b2 base × height area = c2 = 62 + 62 2 c2 = 72 x2 18 = c = √72 2 c = √(36×2) 36 = x2 c = 6√2 6=x 9 10

x

x An isosceles triangle has two equal sides.

A right-angled isosceles triangle has an area of 50. What is the length of the hypotenuse? A right-angled isosceles triangle has an area of 32. What is the length of the hypotenuse?

11 Find the value of x in the following diagram: x

15

5

An isosceles triangle has two equal angles of 45° opposite the equal sides.

9

12 All angles in the following diagrams are either 45° or 90°. Find x. a) b) c) 8 12 16 x

x Chapter 12 Trigonometry 1

x

169

A Couple of Puzzles Exercise 12.13 1 Complete the following: 1 + 3 + 5 = 1 + 3 + 5 + 7 = 1 + 3 + 5 + 7 + 9 =

1 + 3 + 5 + 7 + 9 + 11 = 1 + 3 + 5 + 7 + 9 + 11 + 13 = 1 + 3 + 5 + 7 + ... + 97 + 99 =

A Game Diox is a two player game based on naughts and crosses. The winner is the first person to have three Os or three Xs in a row, column, or diagonal as in the original game of naughts and crosses.

O

Players take turns throwing a die. • an even number means the player must place an X • an odd number means the player must place an O

O

O

X X X

A Sweet Trick The Mobius strip 1 Obtain a long strip of paper that is about 5 cm wide. 2 Make a loop with a half twist and tape the two ends together. 3 Ask your audience what they would expect if you cut the strip of paper in half along the middle of the strip of paper. 4 Cut along the middle of the loop and produce a larger loop and not two loops as would be expected. • What happens if you cut along the middle of the larger loop again? • What happens if you cut a loop with a full twist? Why did the chicken cross the Mobius strip? To get to the same side.

 The Mobius strip is the source of a number of puzzles based on the half twist making the inside surface and the outside surface the same.

170

Try it by drawing a line along the outside that is also the inside.

Investigations Investigation 12.1 Slope The tan ratio is used to measure slope or gradient. 1 What is the angle at which an object begins to slip down the slope (Use the tan ratio to calculate the angle)? 2 Compare this angle with other objects. 3 Why the difference? Investigation 12.2

α

Natural Slope?

Investigate The angle of natural slopes.

160

20

40 50 30 60 150 140 130 120 70 11 0

90

80

10

0

110 70

60 120

50 130

40 140

30 150

10

20

0

16

18

80

0

0

0

10

Use the tan ratio to calculate the height of the tree (What about the height of your eye above the ground?).

0

4

17

Aim the clinometer at the top of the tree and measure the angle of elevation.

10

3

0

Measure a distance out from the base of a tree or flagpole.

18

2

0

Investigation 12.3 Find Heights 1 Make a clinometer using a straw, a protractor, a small weight on the end of a string, and sticky tape.

0

17

0 0

18

170

10

160

20 30

40 50

150 140 130

60

120

0

70 110

0

18

0

10 80

Chapter 12 Trigonometry 1

171

90

10

100 80

70 110

50

60

130

120

40 140

30 150

20

0

16

0

17

Technology

Use a spreadsheet to solve the previous exercises.

opposite adjacent

po

hy

θ

e

us

ten

opposite

tan θ =

adjacent

Technology 12.1 The Tan Ratio and the Spreadsheet a) Given the opposite and adjacent, find the angle. 3 α

4

a Opposite 3

1 2

b Adjacent 4

c Tan α 0.75

d α 36.87

Enter the formula:

=atan(c2)*180/pi() The *180/pi() is needed to convert radians to degrees.

b) Given the angle and adjacent, find the opposite. x

1 2

23° 47

a Angle 23

b Adjacent 47

c Opposite 19.95

Enter the formula:

=tan(a2*pi()/180)*b2 The *pi()/180 is needed to convert degrees to radians.

a) Given the opposite and adjacent, solve the triangle. a Opposite 3.1

1 2

c α a = 3.6

172

b Adjacent 3.6

c α 40.73

d β 49.27

Enter the formula:

β b = 3.1

=atan(a2/b2)*180/pi() The *180/pi() is needed to convert radians to degrees.

e Hypotenuse 4.75 Enter the formula:

=sqrt(a2*a2+b2*b2) Enter the formula:

=180−90−c2

Chapter Review 1 C

Exercise 12.14 1 For the adjacent triangle, name: a) the hypotenuse. b) the side adjacent to the angle. c) the side opposite the angle.

β

A

2 Use Pythagoras' Theorem to find the unknown: a) b) x

20

x

c) 7

29

8.6

3 Find the unknown in each of the following triangles: a) b) c) 5 3 x α 4 32° 71

β

c

93 α

63

5

8.9 x

53°

Solve means 'find all unknowns'.

4 Solve the following triangles: a) b) β

246

x

7.3

15

55

B

c) c

122

39.1

β c

α 76.5

13 m out from the base of a flagpole, a clinometer measures the angle of elevation to the top of the flagpole as 37°. Find the height of the flagpole.

x 13

6

A ship sails due north for 15 km, then on a bearing of 160° until the ship is due east of its starting point. How far is the ship from its starting point?

α

37°

N 0° W 270°

E 90° 180° S

Chapter 12 Trigonometry 1

173

Chapter Review 2 B

Exercise 12.15 1 For the adjacent triangle, name: a) the hypotenuse. b) the side adjacent to the angle. c) the side opposite the angle.

β

C

2 Use Pythagoras' Theorem to find the unknown: a) b) x

28

A

c) 4

50

9.2

x

x

7.6

21

3 Find the unknown in each of the following triangles: a) b) c) 10 α

6

2.1 x

x

8

471

49°

35° 43 Solve means 'find all unknowns'.

4 Solve the following triangles: a) b) 78

β

c

95

133 α

β

c) c 176

44.2

c

α

5

18 m out from the base of a tree, a clinometer measures the angle of elevation to the top of the tree as 52°. Find the height of the tree.

6

A ship sails due East for 60 km, then on a bearing of 225° until the ship is due south of its starting point. How far is the ship from its starting point?

80.3

x 52° 18 N 0° W 270°

E 90° 180° S

174

β

α

Measurement and Geometry  Using units of measurement  Calculate the volume of cylinders and solve related problems.  Solve problems involving the volume of right prisms.  Build on the understanding of volume to become fluent with calculation, and identify that volume relationships are used in the workplace and everyday life.

Paradox, n. 1. a statement appearing to be absurd but containing a truth. 2. a thing showing a contradiction.

A TASK yy Research visual/optical paradoxes/ illusions. yy Make a poster for the classroom.

A LITTLE BIT OF HISTORY Euler (1707-1783), a famous mathematician, has written that artists are quite skilled at using visual illusions in their work. Euler suggested that the whole of the art of painting is based on visual illusions. Ptolemy (90-168), an influential Greek astronomer famous for developing theory from experimental data - the scientific method, was intrigued by the increasing size of celestial bodies as they approach the horizon - The magnificent size of a full moon as it arises in the east.

175

Area Warmup Rectangle

Circle

Triangle

b b

Area = l×b

r

h

l

l

Area = ½bh

b

Area = πr2

Exercise 13.1 Calculate the area of each of the following shapes:

2.3cm

6.2cm

4.6 mm

4.7cm

3.7cm

Area = l×b = 3.7cm × 2.3cm = 8.51 cm2 1

Area = ½bh Area = πr2 = 0.5×4.7cm×6.2cm = π×4.6mm×4.6mm = 14.57 cm2 = 66.48 mm2 2

3

7.1m 7.1m

8m

4

5 8.4km 13.7km

5.3cm

6m

6 12.1m

9.8cm 6.5cm

7

A rectangular paddock is 120 m by 140 m. What is the area of the paddock in square metres and hectares (1 hectare = 10 000m2)?

8

A paddock, in the shape of a triangle, has a base of 230 m and a perpendicular height of 370 m. What is the area of the paddock in square metres and hectares?

9

The sprinkler sprays water in a circle with a radius of 4.5 m. Calculate the area covered by the sprinkler. A hectare is the area of a square 100 m by 100 m.

10 The pizza dish has a diameter of 29 cm. Calculate the area of the pizza dish. The diameter is twice the radius.

11 A classroom has a length of 6 m and a width of 10 m. The floor of the classroom is to be covered with carpet tiles. If each carpet tile has a length of 30 cm and a width of 45 cm, how many carpet tiles are needed to cover the floor?

176

Composite Shapes

Composite shapes can be rectangles, triangles, and circles composed together.

Exercise 13.2 Calculate the shaded area of each of the following composite shapes: 4m

6m

4m

Area = Area of triangle + Area of rectangle + Area of circle = ½bh + l×b + πr2 = ½×4×4 + 6×4 + π×22 = 8 + 24 + 12.57 Area = 44.57 m2 1

2

6m

14 cm 7 cm

9m 5m

19 cm

11 m

3

4

6 mm

70 m

6 mm 5 cm

6

m 5c

5c

m

5

40 m

8m

4.33 cm

m

5c

m

5c 5 cm

7

8

7m

37 mm 5m

Chapter 13 Volume

177

Prisms

The volume is the space occupied by the prism.

A prism has the same cross-section along its length. If the cross-section is a triangle, it is a triangular-based prism If the cross-section is a rectangle, it is a rectangular-based prism If the cross-section is a circle, it is a cylinder.

Units of Volume

÷ 1 000 000 ÷ 1000

1cm3 = 1000mm3 1m3 = 1 000 000cm3

mm

m3

cm3

3

× 1000

× 1 000 000

Exercise 13.3 1 Complete the following unit conversions: 8500 mm3 to cm3 2.3 m3 to cm3 8500 mm3 = 8500 ÷ 1000 cm3 2.3 m3 = 2.3 × 1 000 000 cm3 = 8.5 cm3 = 2 300 000 cm3

a) c) e) g)

÷ 1000

÷ 1000

mL

b) d) f) h)

6700 mm3 to cm3 35 000 000 cm3 to m3 4.9 m3 to cm3 8.3 cm3 to mm3

× 1000

L

× 1000

900 mm3 to cm3 2 400 000 cm3 to m3 1.9 m3 to cm3 0.6 cm3 to mm3

÷ 1000

ML

kL

1L = 1000mL 1kL = 1000L 1ML = 1000kL

× 1000

MegaLitre

2 Complete the following unit conversions: 6500 mL to L 2.8 ML to L 6500 mL = 6500 ÷ 1000 L 2.8 ML = 2.8 × 1 000 000 L = 6.5 L = 2 800 000 L

178

a) 4200 mL to L c) 5800 L to kL e) 7.3 kL to L

b) 61.3 ML to L d) 720 000 L to ML f) 0.61 kL to L

Volume of Prisms

"I'm not a prism," I repeat constantly.

Prisms are three-dimensional shapes that have a constant cross-section.

Exercise 13.4 Find the volume of each of the following prisms:

9m

2cm

4m

6m

r = 3cm

5m

2m 4m

V = Area of base × height V = Area of base × height V = Area of base × height = πr2 × h =l×b×h = ½bh × h = π×32×2 =4×2×9 = ½×5×4×6 V = 56.55 cm3 V = 72 m3 V = 60 m3 1

2 6 cm

7.8 cm

r = 5 cm

r = 3.4 cm

3

4

2.7 m

15 m

7m

1.6

m

5m

5

1.3 m

6

9m

24 m

5m

7m

53 m

67 m

Chapter 13 Volume

179

Composite Solids Exercise 13.5 Calculate the volume of each of the following solids: 13 m

13 m 2.1 m

2.1 m

9m

17 m



2.1 m

9m

4m

13 m

Volume = Volume rectangular prism + Volume triangular based prism = l×b×h + ½bh×h = 13×9×2.1 + ½×4×2.1×9 Volume = 283.5 m3 2 3m

1

3m

6m

18 cm

3m

3m 3m

12 cm

cm

18 cm

4 2.5 m

3

36

10

21 m 2.3 m

3m

m

24 m

4m

6

7m

10 cm

cm

5

24 cm

42

.7

12 cm

18.4 cm

Volume, n. 1. amount of space contained by a three-dimensional object. SI unit is cubic metre (m3). 1 L = 1000 mL 1L = 1000cm3 of water 1L = 1 kg

180

15 cm

Practical Applications Exercise 13.6 What is the cost of concrete for the shed slab? The slab is 7.2 m by 2.4 m by 15 cm. Concrete delivered is $180 per cubic metre. V = Area of base × height =l×b×h = 7.2 × 2.4 × 0.15 {15 cm = 0.15 m} V = 2.59 m3 Thus need 3 cubic metres of concrete Cost = 3×$180 = $540. The concrete for the slab will cost $540 1 2

What is the cost of concrete for the shed slab? The slab is 6.8 m by 3.4 m by 18 cm. Concrete delivered is $205 per cubic metre. What is the cost of concrete for the garden feature? Concrete delivered is $210 per cubic metre.

1.2

0.5 m

m

2.3 m

0.3 m

0.5

m

Estimate the volume of the dam. V = 0.4×Area of surface×depth = 0.4 × l × b × d = 0.4 × 45 × 40 × 4 V = 2880 m3 = 2 880 000 litres The volume is estimated at 2.9 megalitres.

40

45 m

m

0.4 is a correction for the slope of the sides of the dam.

Depth = 4 m

3

Estimate the volume of a dam whose surface approximates a rectangle 67 m by 38 m and with a depth of 5 m (use a correction factor of 0.4 for the slope of the sides of the dam)?

4

The water tank has a diameter of 3.3 m and a height of 2.2 m (Some of the tank is in the ground). How much water will the tank hold?

5

How many litres of water is needed to fill the swimming pool?

1m3 = 1000 L

1.2 m

15

m

35 m

2m

Chapter 13 Volume

181

Mental Computation Exercise 13.7

You need to be a good mental athlete because many everyday problems are solved mentally.

1 Spell Cylinder. 2 Area of a circle formula? 3 Volume of a prism? 4 How many litres in a megalitre? 5 In the triangle, what is tanα? 6 Two sides in a right-angled triangle are 1 and 2. Hypotenuse? 7 Write in scientific notation: 5 000 8 Write in scientific notation: 0.000 2 9 107÷104 10 23×9 23×9

2 α 1 c2=12+22 c2=1+4 c = √5

= 23×(10−1) = 230−23 = 207

Exercise 13.8 1 Spell Rectangular based Prism. 2 Area of a circle formula? 3 Volume of a prism? 4 What is the weight of 1 L of water? 5 In the triangle, what is tanα? 6 Two sides in a right-angled triangle are 1 and 3. Hypotenuse? 7 Write in scientific notation: 80 000 8 Write in scientific notation: 0.000 03 9 107×104 10 34×9

3 α 1

"The volume of a prism is area of base by height," said Jess with some capacity.

Exercise 13.9 1 Spell Triangular based Prism. 2 Area of a circle formula? 3 Volume of a prism? 4 How many litres in a kilolitre? 5 In the triangle, what is tanα? 6 Two sides in a right-angled triangle are 1 and 1. Hypotenuse? 7 Write in scientific notation: 9 000 000 8 Write in scientific notation: 0.000 000 1 9 105÷107 10 46×9

α 1

1

Turn your wounds into wisdom - Oprah Winfrey.

182

Competition Questions Build maths muscle and prepare for mathematics competitions at the same time.

Exercise 13.10 1 60 centicubes are glued together to form a rectangular based prism. If the area of the base is 20 cm2, what is the height of the prism? 2

60 centicubes are glued together to form a rectangular based prism. If the perimeter of the base is 14 cm, what is the height of the prism?

3

If the net is folded to form a cube, which letter is opposite C?

F A

B

D

E

C 5

4

The net is folded to form a cube, If the three numbers at each corner are added, what is the largest sum?

5

A box measures 20 cm by 30 cm by 40 cm. What is the volume, in litres, of the box?

6

A box of baking soda measures 3 cm by 5 cm by 7 cm. (1 teaspoon = 5 cm3). How many teaspoons of baking soda are expected to be in the box?

1

2

3

4

6

25 mm of rain on a flat roof puts 2 500 L of water into the tank. If all of the rain on the roof goes into the tank, what is the area of the roof? Volume = 2 500 L Area of roof × depth = 2 500 x 1000 cm2 Area of roof × 2.5 cm = 2 500 000 cm2 Area of roof =2 500 000 ÷ 2.5 cm2 =1 000 000 cm2 =100 m2

An olympic pool holds about 1 megalitre of water.

7

20 mm of rain on a flat roof puts 2 000 L of water into the tank. If all of the rain on the roof goes into the tank, what is the area of the roof?

8

A 4 metre square-based tank has water to a depth of 4 metres. If a cube of side 2 metres is placed in the tank, what is then the level of water in the tank?

9

A 6 metre square-based tank has water to a depth of 4 metres. If a cube of side 2 metres is placed in the tank, what is then the level of water in the tank? Chapter 13 Volume

183

Investigations Investigation 13.1

Nets of Cubes

Which of the following nets will fold to form a cube? Make copies and try them.

Investigate How many different nets can be folded to make a cube?

Investigation 13.2 How big is 1 cubic centimetre (cm3)? To become familiar with the volume of a cubic centimetre, make a 1 cm by 1 cm by 1 cm box or use centicubes. Use the cubic centimetre to estimate volumes in your classroom: • The volume of a calculator. If you have centicubes, • the volume of a pencil case. use the centicubes to • etc. make a 5×5×5 box and How close were your estimates to the actual volumes? show that the volume is 125 cm3.

Investigation 13.3 How big is 1 cubic metre (m3)? To become familiar with the space of a cubic metre, make a 1 m by 1 m by 1 m frame Use the cubic metre to estimate volumes in your classroom: • The volume of a desk. • The volume of the classroom. How close were your estimates to the actual volumes?

Investigation 13.4

The Soma cube

Investigate The Soma cube

184

A cubic metre of water weights 1 tonne (1000L).

A Couple of Puzzles

x

x+1

Exercise 13.11 1 When a book is open the sum of the two pages is 15. What is the number of the next page.

x+x+1=15

2

A drum is full. When 40 L is taken out of the drum, the drum is one-third full. How much will the drum hold?

3

How many coins, of the same size, are needed to ‘ring’ a coin?

A Game Dart maths. The first to get a total of exactly 301 is the winner (two to five players). 1 When it is your turn, throw a dice three times. 2 Then try to use ( ), +, –, ×, or ÷ with your three numbers to progress towards the exact total of 301.

A Sweet Trick 1 2 3

Ask your audience to draw a 3×3 box around any 9 numbers on a calendar You tell them the sum is 144 They take ages to check the sum on their calculator. You just multiply the centre number by 9. 16×9 = 144.

5(2+1) = 125

S

M

6 13 20 27

7 14 21 28

T 1 8 15 22 29

W 2 9 16 23 30

T 3 10 17 24 31

F 4 11 18 25

S 5 12 19 26

One way to multiply by 9: Multiply by 10 then subtract the number. 16×9 = 16×10−16 = 144

Chapter 13 Volume

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Technology Technology 13.1 Drawing cubes )) Use a search phrase such as "drawing cubes" or "drawing cubes perspective" to find online tutorials for drawing cubes. )) When you feel that you are good at drawing cubes, learn to draw cylinders. )) Now try your hand at drawing triangular based prisms and other prisms..

Technology 13.2 Drawing Prisms )) Interactive geometry software or dynamic geometry environments are about making and manipulating geometric objects. )) Use a search phrase such as "interactive geometry software" to find a list of such software. )) Experiment with one or two of them.

Technology 13.3 Applets )) Use a search phrase such as 'volume applet' or 'interactive volume' to find one of the many applets on the Internet. Experiment with them. )) Use a search phrase such as 'volume calculator' to find one of the many volume calculators on the Internet. Experiment with them. )) Use a search phrase such as "java soma cube" to find an applet that lets you experiment with the Soma cube.

Aerospace Engineers design, develop, manufacture, and maintain aeroplanes, helicopters, spacecraft, missiles, etc. • Relevant school subjects are English, Mathematics, Physics, and Chemistry. • Courses generally involves a University engineering degree.

186

Chapter Review 1 Exercise 13.12 1 Calculate the area of each of the following shapes: a) b)

c)

4.8cm

6m

8m

13 m 7m

7m

15 m

2

Make the following unit conversions: a) 7200 mm3 to cm3 c) 2.7 m3 to cm3 e) 4600 mL to L g) 1.1 kL to L

3

Calculate the volume of each of the following solids:

a)

b) d) f) h)

58 000 000 cm3 to m3 0.9 cm3 to mm3 410 000 L to kL 0.25 L to mL

b)

9 cm

19 m

8m

r = 4 cm

d)

3m

c)

10

6m 8 cm

8 cm

3.3 m

m

5m 10 cm

4

What is the cost of concrete for the shed slab? The slab is 6.4 m by 3.8 m by 15 cm. Concrete delivered is $190 per cubic metre

5

How many litres of water is needed to fill the swimming pool?

1.2 m

15

m

35 m

1.8 m

Chapter 13 Volume

187

Chapter Review 2 Exercise 13.13 1 Calculate the area of each of the following shapes: a) b)

7 mm

7 mm

8.3cm

c)

6.4 cm

6.2cm

2

Make the following unit conversions: a) 92 000 mm3 to cm3 c) 9.8 m3 to cm3 e) 5600 mL to L g) 4.1 kL to L

3

Calculate the volume of each of the following solids:

a)

b) d) f) h)

b)

5.7 m 2.5

9 800 000 cm3 to m3 0.6 cm3 to mm3 350 000 L to kL 0.49 L to mL

m

5m

9m

7m

3.4 m

c)

d)

13 cm

26 m 3.4 m 32 m

12 cm

6m 15 cm

4

What is the cost of concrete for the shed slab? The slab is 9.2 m by 3.1 m by 21 cm. Concrete delivered is $180 per cubic metre

5

188

How many litres of water is needed to fill the swimming pool?

12

m

50 m

2.2 m

1.1 m

Statistics and Probability  Chance  List all outcomes for two-step chance experiments, both with and without replacement using tree diagrams or arrays.  Assign probabilities to outcomes and determine probabilities for events.  Calculate relative frequencies from given or collected data to estimate probabilities of events involving ‘and’ or ‘or’.  Posing ‘and’, ‘or’, ‘not’ and ‘given’ probability questions about objects or people.  Collect data to answer the questions using Venn diagrams or two-way tables.

A TASK

Is it OK for a family to spend $50 each week playing Gold Lotto?

Organise a class debate: Two teams, three speakers per team, debate the topic. A toss of a coin usually decides which team, Affirmative, agrees with the topic, and which team, Negative, disagrees with the topic. Four minutes is normally given to each speaker with a warning bell at the end of three minutes and two bells at four minutes. The teams speak in the following order: 1 First Affirmative 2 First Negative 3 Second Affirmative 4 Second Negative 5 Third Affirmative 6 Third Negative

Define the topic. Introduce the team’s main argument. Rebut the opening main argument 'why it isn't true.' Further support of team's argument. Some rebuttal. Rebuttal of affirmative's argument. Rebuttal of opposition. Summary of team's argument. Convincing concluding statement.

A LITTLE BIT OF HISTORY Sir Francis Bacon (1561-1626), the father of deductive reasoning, was the first to use inductive thinking as a basis for scientific procedure. Thomas Bayes (1702-1761) provided the first mathematical basis for inductive reasoning and developed Bayesian probability theory.

Sir Francis Bacon

189

Warm Up



Probability is the chance of something happening.

impossible

fifty-fifty

0

certain

0.5

1

Exercise 14.1 1 Copy the above probability scale and add each of the following to the scale. a) The day after Friday will be Saturday. b) We have Buckley's chance of winning the match. c) Throw a coin and it will show a tail. d) You will use a computer today. e) We have a 3 in 4 chance of picking the winner. 1

f) The probability of winning first prize is 8000000 or 0.000 000 125.



Pr obability =

Number of possible outcomes Total number of outcomes

A standard 6-sided die is thrown. What is the probability of each of the following happening? a) 5 b) odd c) 6?

Sample space = {1,2,3,4,5,6}

Sample space = {1,2,3,4,5,6,7,8}

1

1

a) P(5) = 6

{there is one 5} a) P(2) = 8 3

1

b) P(odd) = 6 = 2 {1,3,5 are odd} 2

1

c) P(6}

2

A six-sided die is thrown. a) 3 d) not even g) >6

3

A bag contains five blue marbles and three red marbles. What is the probability of taking a marble from the bag that is: a) red? b) blue? c) not red? d) not blue? e) blue or red? f) white?

What is the probability of each of the following? b) even c) odd e)